fmpz_poly.h – univariate polynomials over the integers

Introduction

The fmpz_poly_t data type represents elements of \(\mathbb{Z}[x]\). The fmpz_poly module provides routines for memory management, basic arithmetic, and conversions from or to other types.

Each coefficient of an fmpz_poly_t is an integer of the FLINT fmpz_t type. There are two advantages of this model. Firstly, the fmpz_t type is memory managed, so the user can manipulate individual coefficients of a polynomial without having to deal with tedious memory management. Secondly, a coefficient of an fmpz_poly_t can be changed without changing the size of any of the other coefficients.

Unless otherwise specified, all functions in this section permit aliasing between their input arguments and between their input and output arguments.

Simple example

The following example computes the square of the polynomial \(5x^3 - 1\).

#include "fmpz_poly.h"

int main()
{
    fmpz_poly_t x, y;
    fmpz_poly_init(x);
    fmpz_poly_init(y);
    fmpz_poly_set_coeff_ui(x, 3, 5);
    fmpz_poly_set_coeff_si(x, 0, -1);
    fmpz_poly_mul(y, x, x);
    fmpz_poly_print(x); flint_printf("\n");
    fmpz_poly_print(y); flint_printf("\n");
    fmpz_poly_clear(x);
    fmpz_poly_clear(y);
}

The output is:

4  -1 0 0 5
7  1 0 0 -10 0 0 25

Definition of the fmpz_poly_t type

The fmpz_poly_t type is a typedef for an array of length 1 of fmpz_poly_struct’s. This permits passing parameters of type fmpz_poly_t by reference in a manner similar to the way GMP integers of type mpz_t can be passed by reference.

In reality one never deals directly with the struct and simply deals with objects of type fmpz_poly_t. For simplicity we will think of an fmpz_poly_t as a struct, though in practice to access fields of this struct, one needs to dereference first, e.g. to access the length field of an fmpz_poly_t called poly1 one writes poly1->length.

An fmpz_poly_t is said to be normalised if either length is zero, or if the leading coefficient of the polynomial is non-zero. All fmpz_poly functions expect their inputs to be normalised, and unless otherwise specified they produce output that is normalised.

It is recommended that users do not access the fields of an fmpz_poly_t or its coefficient data directly, but make use of the functions designed for this purpose, detailed below.

Functions in fmpz_poly do all the memory management for the user. One does not need to specify the maximum length or number of limbs per coefficient in advance before using a polynomial object. FLINT reallocates space automatically as the computation proceeds, if more space is required. Each coefficient is also managed separately, being resized as needed, independently of the other coefficients.

Types, macros and constants

type fmpz_poly_struct

type fmpz_poly_t

Memory management

void fmpz_poly_init(fmpz_poly_t poly)

Initialises poly for use, setting its length to zero. A corresponding call to fmpz_poly_clear() must be made after finishing with the fmpz_poly_t to free the memory used by the polynomial.

void fmpz_poly_init2(fmpz_poly_t poly, slong alloc)

Initialises poly with space for at least alloc coefficients and sets the length to zero. The allocated coefficients are all set to zero.

void fmpz_poly_realloc(fmpz_poly_t poly, slong alloc)

Reallocates the given polynomial to have space for alloc coefficients. If alloc is zero the polynomial is cleared and then reinitialised. If the current length is greater than alloc the polynomial is first truncated to length alloc.

void fmpz_poly_fit_length(fmpz_poly_t poly, slong len)

If len is greater than the number of coefficients currently allocated, then the polynomial is reallocated to have space for at least len coefficients. No data is lost when calling this function.

The function efficiently deals with the case where fit_length is called many times in small increments by at least doubling the number of allocated coefficients when length is larger than the number of coefficients currently allocated.

void fmpz_poly_clear(fmpz_poly_t poly)

Clears the given polynomial, releasing any memory used. It must be reinitialised in order to be used again.

void _fmpz_poly_normalise(fmpz_poly_t poly)

Sets the length of poly so that the top coefficient is non-zero. If all coefficients are zero, the length is set to zero. This function is mainly used internally, as all functions guarantee normalisation.

void _fmpz_poly_set_length(fmpz_poly_t poly, slong newlen)

Demotes the coefficients of poly beyond newlen and sets the length of poly to newlen.

void fmpz_poly_attach_truncate(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)

This function sets the uninitialised polynomial trunc to the low \(n\) coefficients of poly, or to poly if the latter doesn’t have \(n\) coefficients. The polynomial trunc not be cleared or used as the output of any Flint functions.

void fmpz_poly_attach_shift(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)

This function sets the uninitialised polynomial trunc to the high coefficients of poly, i.e. the coefficients not among the low \(n\) coefficients of poly. If the latter doesn’t have \(n\) coefficients trunc is set to the zero polynomial. The polynomial trunc not be cleared or used as the output of any Flint functions.

Polynomial parameters

slong fmpz_poly_length(const fmpz_poly_t poly)

Returns the length of poly. The zero polynomial has length zero.

slong fmpz_poly_degree(const fmpz_poly_t poly)

Returns the degree of poly, which is one less than its length.

Assignment and basic manipulation

void fmpz_poly_set(fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets poly1 to equal poly2.

void fmpz_poly_set_si(fmpz_poly_t poly, slong c)

Sets poly to the signed integer c.

void fmpz_poly_set_ui(fmpz_poly_t poly, ulong c)

Sets poly to the unsigned integer c.

void fmpz_poly_set_fmpz(fmpz_poly_t poly, const fmpz_t c)

Sets poly to the integer c.

int _fmpz_poly_set_str(fmpz *poly, const char *str)

Sets poly to the polynomial encoded in the null-terminated string str. Assumes that poly is allocated as a sufficiently large array suitable for the number of coefficients present in str.

Returns \(0\) if no error occurred. Otherwise, returns a non-zero value, in which case the resulting value of poly is undefined. If str is not null-terminated, calling this method might result in a segmentation fault.

int fmpz_poly_set_str(fmpz_poly_t poly, const char *str)

Imports a polynomial from a null-terminated string. If the string str represents a valid polynomial returns \(0\), otherwise returns \(1\).

Returns \(0\) if no error occurred. Otherwise, returns a non-zero value, in which case the resulting value of poly is undefined. If str is not null-terminated, calling this method might result in a segmentation fault.

char *_fmpz_poly_get_str(const fmpz *poly, slong len)

Returns the plain FLINT string representation of the polynomial (poly, len).

char *fmpz_poly_get_str(const fmpz_poly_t poly)

Returns the plain FLINT string representation of the polynomial poly.

char *_fmpz_poly_get_str_pretty(const fmpz *poly, slong len, const char *x)

Returns a pretty representation of the polynomial (poly, len) using the null-terminated string x as the variable name.

char *fmpz_poly_get_str_pretty(const fmpz_poly_t poly, const char *x)

Returns a pretty representation of the polynomial poly using the null-terminated string x as the variable name.

void fmpz_poly_zero(fmpz_poly_t poly)

Sets poly to the zero polynomial.

void fmpz_poly_one(fmpz_poly_t poly)

Sets poly to the constant polynomial one.

void fmpz_poly_zero_coeffs(fmpz_poly_t poly, slong i, slong j)

Sets the coefficients of \(x^i, \dotsc, x^{j-1}\) to zero.

void fmpz_poly_swap(fmpz_poly_t poly1, fmpz_poly_t poly2)

Swaps poly1 and poly2. This is done efficiently without copying data by swapping pointers, etc.

void _fmpz_poly_reverse(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, n) to the reverse of (poly, n), where poly is in fact an array of length len. Assumes that 0 < len <= n. Supports aliasing of res and poly, but the behaviour is undefined in case of partial overlap.

void fmpz_poly_reverse(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

This function considers the polynomial poly to be of length \(n\), notionally truncating and zero padding if required, and reverses the result. Since the function normalises its result res may be of length less than \(n\).

void fmpz_poly_truncate(fmpz_poly_t poly, slong newlen)

If the current length of poly is greater than newlen, it is truncated to have the given length. Discarded coefficients are not necessarily set to zero.

void fmpz_poly_set_trunc(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to a copy of poly, truncated to length n.

Randomisation

void fmpz_poly_randtest(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

Sets \(f\) to a random polynomial with up to the given length and where each coefficient has up to the given number of bits. The coefficients are signed randomly.

void fmpz_poly_randtest_unsigned(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

Sets \(f\) to a random polynomial with up to the given length and where each coefficient has up to the given number of bits.

void fmpz_poly_randtest_not_zero(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

As for fmpz_poly_randtest() except that len and bits may not be zero and the polynomial generated is guaranteed not to be the zero polynomial.

void fmpz_poly_randtest_no_real_root(fmpz_poly_t p, flint_rand_t state, slong len, flint_bitcnt_t bits)

Sets p to a random polynomial without any real root, whose length is up to len and where each coefficient has up to the given number of bits.

void fmpz_poly_randtest_irreducible1(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)

void fmpz_poly_randtest_irreducible2(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)

void fmpz_poly_randtest_irreducible(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)

Sets p to a random irreducible polynomial, whose length is up to len and where each coefficient has up to the given number of bits. There are two algorithms: irreducible1 generates an irreducible polynomial modulo a random prime number and lifts it to the integers; irreducible2 generates a random integer polynomial, factors it, and returns a random factor. The default function chooses randomly between these methods.

Getting and setting coefficients

void fmpz_poly_get_coeff_fmpz(fmpz_t x, const fmpz_poly_t poly, slong n)

Sets \(x\) to the \(n\)-th coefficient of poly. Coefficient numbering is from zero and if \(n\) is set to a value beyond the end of the polynomial, zero is returned.

slong fmpz_poly_get_coeff_si(const fmpz_poly_t poly, slong n)

Returns coefficient \(n\) of poly as a slong. The result is undefined if the value does not fit into a slong. Coefficient numbering is from zero and if \(n\) is set to a value beyond the end of the polynomial, zero is returned.

ulong fmpz_poly_get_coeff_ui(const fmpz_poly_t poly, slong n)

Returns coefficient \(n\) of poly as a ulong. The result is undefined if the value does not fit into a ulong. Coefficient numbering is from zero and if \(n\) is set to a value beyond the end of the polynomial, zero is returned.

fmpz *fmpz_poly_get_coeff_ptr(const fmpz_poly_t poly, slong n)

Returns a reference to the coefficient of \(x^n\) in the polynomial, as an fmpz *. This function is provided so that individual coefficients can be accessed and operated on by functions in the fmpz module. This function does not make a copy of the data, but returns a reference to the actual coefficient.

Returns NULL when \(n\) exceeds the degree of the polynomial.

This function is implemented as a macro.

fmpz *fmpz_poly_lead(const fmpz_poly_t poly)

Returns a reference to the leading coefficient of the polynomial, as an fmpz *. This function is provided so that the leading coefficient can be easily accessed and operated on by functions in the fmpz module. This function does not make a copy of the data, but returns a reference to the actual coefficient.

Returns NULL when the polynomial is zero.

This function is implemented as a macro.

void fmpz_poly_set_coeff_fmpz(fmpz_poly_t poly, slong n, const fmpz_t x)

Sets coefficient \(n\) of poly to the fmpz value x. Coefficient numbering starts from zero and if \(n\) is beyond the current length of poly then the polynomial is extended and zero coefficients inserted if necessary.

void fmpz_poly_set_coeff_si(fmpz_poly_t poly, slong n, slong x)

Sets coefficient \(n\) of poly to the slong value x. Coefficient numbering starts from zero and if \(n\) is beyond the current length of poly then the polynomial is extended and zero coefficients inserted if necessary.

void fmpz_poly_set_coeff_ui(fmpz_poly_t poly, slong n, ulong x)

Sets coefficient \(n\) of poly to the ulong value x. Coefficient numbering starts from zero and if \(n\) is beyond the current length of poly then the polynomial is extended and zero coefficients inserted if necessary.

Comparison

int fmpz_poly_equal(const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Returns \(1\) if poly1 is equal to poly2, otherwise returns \(0\). The polynomials are assumed to be normalised.

int fmpz_poly_equal_trunc(const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Return \(1\) if poly1 and poly2, notionally truncated to length \(n\) are equal, otherwise return \(0\).

int fmpz_poly_is_zero(const fmpz_poly_t poly)

Returns \(1\) if the polynomial is zero and \(0\) otherwise.

This function is implemented as a macro.

int fmpz_poly_is_one(const fmpz_poly_t poly)

Returns \(1\) if the polynomial is one and \(0\) otherwise.

int fmpz_poly_is_unit(const fmpz_poly_t poly)

Returns \(1\) if the polynomial is the constant polynomial \(\pm 1\), and \(0\) otherwise.

int fmpz_poly_is_gen(const fmpz_poly_t poly)

Returns \(1\) if the polynomial is the degree \(1\) polynomial \(x\), and \(0\) otherwise.

Addition and subtraction

void _fmpz_poly_add(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the sum of (poly1, len1) and (poly2, len2). It is assumed that res has sufficient space for the longer of the two polynomials.

void fmpz_poly_add(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the sum of poly1 and poly2.

void fmpz_poly_add_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Notionally truncate poly1 and poly2 to length \(n\) and then set res to the sum.

void _fmpz_poly_sub(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to (poly1, len1) minus (poly2, len2). It is assumed that res has sufficient space for the longer of the two polynomials.

void fmpz_poly_sub(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to poly1 minus poly2.

void fmpz_poly_sub_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Notionally truncate poly1 and poly2 to length \(n\) and then set res to the sum.

void fmpz_poly_neg(fmpz_poly_t res, const fmpz_poly_t poly)

Sets res to -poly.

Scalar absolute value, multiplication and division

void fmpz_poly_scalar_abs(fmpz_poly_t res, const fmpz_poly_t poly)

Sets poly1 to the polynomial whose coefficients are the absolute value of those of poly2.

void fmpz_poly_scalar_mul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly2 times \(x\).

void fmpz_poly_scalar_mul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

Sets poly1 to poly2 times the signed slong x.

void fmpz_poly_scalar_mul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 times the ulong x.

void fmpz_poly_scalar_mul_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong exp)

Sets poly1 to poly2 times 2^exp.

void fmpz_poly_scalar_addmul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

void fmpz_poly_scalar_addmul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

void fmpz_poly_scalar_addmul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly1 + x * poly2.

void fmpz_poly_scalar_submul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly1 - x * poly2.

void fmpz_poly_scalar_fdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly2 divided by the fmpz_t x, rounding coefficients down toward \(- \infty\).

void fmpz_poly_scalar_fdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

Sets poly1 to poly2 divided by the slong x, rounding coefficients down toward \(- \infty\).

void fmpz_poly_scalar_fdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 divided by the ulong x, rounding coefficients down toward \(- \infty\).

void fmpz_poly_scalar_fdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 divided by 2^x, rounding coefficients down toward \(- \infty\).

void fmpz_poly_scalar_tdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly2 divided by the fmpz_t x, rounding coefficients toward \(0\).

void fmpz_poly_scalar_tdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

Sets poly1 to poly2 divided by the slong x, rounding coefficients toward \(0\).

void fmpz_poly_scalar_tdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 divided by the ulong x, rounding coefficients toward \(0\).

void fmpz_poly_scalar_tdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 divided by 2^x, rounding coefficients toward \(0\).

void fmpz_poly_scalar_divexact_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

Sets poly1 to poly2 divided by the fmpz_t x, assuming the division is exact for every coefficient.

void fmpz_poly_scalar_divexact_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

Sets poly1 to poly2 divided by the slong x, assuming the coefficient is exact for every coefficient.

void fmpz_poly_scalar_divexact_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

Sets poly1 to poly2 divided by the ulong x, assuming the coefficient is exact for every coefficient.

void fmpz_poly_scalar_mod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)

Sets poly1 to poly2, reducing each coefficient modulo \(p > 0\).

void fmpz_poly_scalar_smod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)

Sets poly1 to poly2, symmetrically reducing each coefficient modulo \(p > 0\), that is, choosing the unique representative in the interval \((-p/2, p/2]\).

slong _fmpz_poly_remove_content_2exp(fmpz *pol, slong len)

Remove the 2-content of pol and return the number \(k\) that is the maximal non-negative integer so that \(2^k\) divides all coefficients of the polynomial. For the zero polynomial, \(0\) is returned.

void _fmpz_poly_scale_2exp(fmpz *pol, slong len, slong k)

Scale (pol, len) to \(p(2^k X)\) in-place and divide by the 2-content (so that the gcd of coefficients is odd). If k is negative the polynomial is multiplied by \(2^{kd}\).

Bit packing

void _fmpz_poly_bit_pack(nn_ptr arr, const fmpz *poly, slong len, flint_bitcnt_t bit_size, int negate)

Packs the coefficients of poly into bitfields of the given bit_size, negating the coefficients before packing if negate is set to \(-1\).

int _fmpz_poly_bit_unpack(fmpz *poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size, int negate)

Unpacks the polynomial of given length from the array as packed into fields of the given bit_size, finally negating the coefficients if negate is set to \(-1\). Returns borrow, which is nonzero if a leading term with coefficient \(\pm1\) should be added at position len of poly.

void _fmpz_poly_bit_unpack_unsigned(fmpz *poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size)

Unpacks the polynomial of given length from the array as packed into fields of the given bit_size. The coefficients are assumed to be unsigned.

void fmpz_poly_bit_pack(fmpz_t f, const fmpz_poly_t poly, flint_bitcnt_t bit_size)

Packs poly into bitfields of size bit_size, writing the result to f. The sign of f will be the same as that of the leading coefficient of poly.

void fmpz_poly_bit_unpack(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)

Unpacks the polynomial with signed coefficients packed into fields of size bit_size as represented by the integer f.

void fmpz_poly_bit_unpack_unsigned(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)

Unpacks the polynomial with unsigned coefficients packed into fields of size bit_size as represented by the integer f. It is required that f is nonnegative.

Multiplication

void _fmpz_poly_mul_classical(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets (res, len1 + len2 - 1) to the product of (poly1, len1) and (poly2, len2).

Assumes len1 and len2 are positive. Allows zero-padding of the two input polynomials. No aliasing of inputs with outputs is allowed.

void fmpz_poly_mul_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the product of poly1 and poly2, computed using the classical or schoolbook method.

void _fmpz_poly_mullow_classical(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets (res, n) to the first \(n\) coefficients of (poly1, len1) multiplied by (poly2, len2).

Assumes 0 < n <= len1 + len2 - 1. Assumes neither len1 nor len2 is zero.

void fmpz_poly_mullow_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the first \(n\) coefficients of poly1 * poly2.

void _fmpz_poly_mulhigh_classical(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong start)

Sets the first start coefficients of res to zero and the remainder to the corresponding coefficients of (poly1, len1) * (poly2, len2).

Assumes start <= len1 + len2 - 1. Assumes neither len1 nor len2 is zero.

void fmpz_poly_mulhigh_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong start)

Sets the first start coefficients of res to zero and the remainder to the corresponding coefficients of the product of poly1 and poly2.

void _fmpz_poly_mulmid_classical(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the middle len1 - len2 + 1 coefficients of the product of (poly1, len1) and (poly2, len2), i.e. the coefficients from degree len2 - 1 to len1 - 1 inclusive. Assumes that len1 >= len2 > 0.

void fmpz_poly_mulmid_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the middle len(poly1) - len(poly2) + 1 coefficients of poly1 * poly2, i.e. the coefficient from degree len2 - 1 to len1 - 1 inclusive. Assumes that len1 >= len2.

void _fmpz_poly_mul_karatsuba(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets (res, len1 + len2 - 1) to the product of (poly1, len1) and (poly2, len2). Assumes len1 >= len2 > 0. Allows zero-padding of the two input polynomials. No aliasing of inputs with outputs is allowed.

void fmpz_poly_mul_karatsuba(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the product of poly1 and poly2.

void _fmpz_poly_mullow_karatsuba_n(fmpz *res, const fmpz *poly1, const fmpz *poly2, slong n)

Sets res to the product of poly1 and poly2 and truncates to the given length. It is assumed that poly1 and poly2 are precisely the given length, possibly zero padded. Assumes \(n\) is not zero.

void fmpz_poly_mullow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the product of poly1 and poly2 and truncates to the given length.

void _fmpz_poly_mulhigh_karatsuba_n(fmpz *res, const fmpz *poly1, const fmpz *poly2, slong len)

Sets res to the product of poly1 and poly2 and truncates at the top to the given length. The first len - 1 coefficients are set to zero. It is assumed that poly1 and poly2 are precisely the given length, possibly zero padded. Assumes len is not zero.

void fmpz_poly_mulhigh_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong len)

Sets the first len - 1 coefficients of the result to zero and the remaining coefficients to the corresponding coefficients of the product of poly1 and poly2. Assumes poly1 and poly2 are at most of the given length.

void _fmpz_poly_mul_KS(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets (res, len1 + len2 - 1) to the product of (poly1, len1) and (poly2, len2).

Places no assumptions on len1 and len2. Allows zero-padding of the two input polynomials. Supports aliasing of inputs and outputs.

void fmpz_poly_mul_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the product of poly1 and poly2.

void _fmpz_poly_mullow_KS(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets (res, n) to the lowest \(n\) coefficients of the product of (poly1, len1) and (poly2, len2).

Assumes that len1 and len2 are positive, but does allow for the polynomials to be zero-padded. The polynomials may be zero, too. Assumes \(n\) is positive. Supports aliasing between res, poly1 and poly2.

void fmpz_poly_mullow_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the lowest \(n\) coefficients of the product of poly1 and poly2.

void _fmpz_poly_mul_SS(fmpz *output, const fmpz *input1, slong length1, const fmpz *input2, slong length2)

Sets (output, length1 + length2 - 1) to the product of (input1, length1) and (input2, length2).

We must have len1 > 1 and len2 > 1. Allows zero-padding of the two input polynomials. Supports aliasing of inputs and outputs.

void fmpz_poly_mul_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the product of poly1 and poly2. Uses the Schönhage-Strassen algorithm.

void _fmpz_poly_mullow_SS(fmpz *output, const fmpz *input1, slong length1, const fmpz *input2, slong length2, slong n)

Sets (res, n) to the lowest \(n\) coefficients of the product of (poly1, len1) and (poly2, len2).

Assumes that len1 and len2 are positive, but does allow for the polynomials to be zero-padded. We must have len1 > 1 and len2 > 1. Assumes \(n\) is positive. Supports aliasing between res, poly1 and poly2.

void fmpz_poly_mullow_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the lowest \(n\) coefficients of the product of poly1 and poly2.

void _fmpz_poly_mul(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets (res, len1 + len2 - 1) to the product of (poly1, len1) and (poly2, len2). Assumes len1 >= len2 > 0. Allows zero-padding of the two input polynomials. Does not support aliasing between the inputs and the output.

void fmpz_poly_mul(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the product of poly1 and poly2. Chooses an optimal algorithm from the choices above.

void _fmpz_poly_mullow(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets (res, n) to the lowest \(n\) coefficients of the product of (poly1, len1) and (poly2, len2).

Assumes len1 >= len2 > 0 and 0 < n <= len1 + len2 - 1. Allows for zero-padding in the inputs. Does not support aliasing between the inputs and the output.

void fmpz_poly_mullow(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the lowest \(n\) coefficients of the product of poly1 and poly2.

void fmpz_poly_mulhigh_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets the high \(n\) coefficients of res to the high \(n\) coefficients of the product of poly1 and poly2, assuming the latter are precisely \(n\) coefficients in length, zero padded if necessary. The remaining \(n - 1\) coefficients may be arbitrary.

void _fmpz_poly_mulhigh(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong start)

Sets all but the low \(n\) coefficients of \(res\) to the corresponding coefficients of the product of \(poly1\) of length \(len1\) and \(poly2\) of length \(len2\), the remaining coefficients being arbitrary. It is assumed that \(len1 >= len2 > 0\) and that \(0 < n < len1 + len2 - 1\). Aliasing of inputs is not permitted.

FFT precached multiplication

void fmpz_poly_mul_SS_precache_init(fmpz_poly_mul_precache_t pre, slong len1, slong bits1, const fmpz_poly_t poly2)

Precompute the FFT of poly2 to enable repeated multiplication of poly2 by polynomials whose length does not exceed len1 and whose number of bits per coefficient does not exceed bits1.

The value bits1 may be negative, i.e. it may be the result of calling fmpz_poly_max_bits. The function only considers the absolute value of bits1.

Suppose len2 is the length of poly2 and len = len1 + len2 - 1 is the maximum output length of a polynomial multiplication using pre. Then internally len is rounded up to a power of two, \(2^n\) say. The truncated FFT algorithm is used to smooth performance but note that it can only do this in the range \((2^{n-1}, 2^n]\). Therefore, it may be more efficient to recompute \(pre\) for cases where the output length will fall below \(2^{n-1} + 1\). Otherwise the implementation will zero pad them up to that length.

Note that the Schoenhage-Strassen algorithm is only efficient for polynomials with relatively large coefficients relative to the length of the polynomials.

Also note that there are no restrictions on the polynomials. In particular the polynomial whose FFT is being precached does not have to be either longer or shorter than the polynomials it is to be multiplied by.

void fmpz_poly_mul_precache_clear(fmpz_poly_mul_precache_t pre)

Clear the space allocated by fmpz_poly_mul_SS_precache_init.

void _fmpz_poly_mullow_SS_precache(fmpz *output, const fmpz *input1, slong len1, fmpz_poly_mul_precache_t pre, slong trunc)

Write into output the first trunc coefficients of the polynomial (input1, len1) by the polynomial whose FFT was precached by fmpz_poly_mul_SS_precache_init and stored in pre.

For performance reasons it is recommended that all polynomials be truncated to at most trunc coefficients if possible.

void fmpz_poly_mullow_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre, slong n)

Set res to the product of poly1 by the polynomial whose FFT was precached by fmpz_poly_mul_SS_precache_init (and stored in pre). The result is truncated to \(n\) coefficients (and normalised).

There are no restrictions on the length of poly1 other than those given in the call to fmpz_poly_mul_SS_precache_init.

void fmpz_poly_mul_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre)

Set res to the product of poly1 by the polynomial whose FFT was precached by fmpz_poly_mul_SS_precache_init (and stored in pre).

There are no restrictions on the length of poly1 other than those given in the call to fmpz_poly_mul_SS_precache_init.

Squaring

void _fmpz_poly_sqr_KS(fmpz *rop, const fmpz *op, slong len)

Sets (rop, 2*len - 1) to the square of (op, len), assuming that len > 0.

Supports zero-padding in (op, len). Does not support aliasing.

void fmpz_poly_sqr_KS(fmpz_poly_t rop, const fmpz_poly_t op)

Sets rop to the square of the polynomial op using Kronecker segmentation.

void _fmpz_poly_sqr_karatsuba(fmpz *rop, const fmpz *op, slong len)

Sets (rop, 2*len - 1) to the square of (op, len), assuming that len > 0.

Supports zero-padding in (op, len). Does not support aliasing.

void fmpz_poly_sqr_karatsuba(fmpz_poly_t rop, const fmpz_poly_t op)

Sets rop to the square of the polynomial op using the Karatsuba multiplication algorithm.

void _fmpz_poly_sqr_classical(fmpz *rop, const fmpz *op, slong len)

Sets (rop, 2*len - 1) to the square of (op, len), assuming that len > 0.

Supports zero-padding in (op, len). Does not support aliasing.

void fmpz_poly_sqr_classical(fmpz_poly_t rop, const fmpz_poly_t op)

Sets rop to the square of the polynomial op using the classical or schoolbook method.

void _fmpz_poly_sqr(fmpz *rop, const fmpz *op, slong len)

Sets (rop, 2*len - 1) to the square of (op, len), assuming that len > 0.

Supports zero-padding in (op, len). Does not support aliasing.

void fmpz_poly_sqr(fmpz_poly_t rop, const fmpz_poly_t op)

Sets rop to the square of the polynomial op.

void _fmpz_poly_sqrlow_KS(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, n) to the lowest \(n\) coefficients of the square of (poly, len).

Assumes that len is positive, but does allow for the polynomial to be zero-padded. The polynomial may be zero, too. Assumes \(n\) is positive. Supports aliasing between res and poly.

void fmpz_poly_sqrlow_KS(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to the lowest \(n\) coefficients of the square of poly.

void _fmpz_poly_sqrlow_karatsuba_n(fmpz *res, const fmpz *poly, slong n)

Sets (res, n) to the square of (poly, n) truncated to length \(n\), which is assumed to be positive. Allows for poly to be zero-padded.

void fmpz_poly_sqrlow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to the square of poly and truncates to the given length.

void _fmpz_poly_sqrlow_classical(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, n) to the first \(n\) coefficients of the square of (poly, len).

Assumes that 0 < n <= 2 * len - 1.

void fmpz_poly_sqrlow_classical(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to the first \(n\) coefficients of the square of poly.

void _fmpz_poly_sqrlow(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, n) to the lowest \(n\) coefficients of the square of (poly, len).

Assumes len1 >= len2 > 0 and 0 < n <= 2 * len - 1. Allows for zero-padding in the input. Does not support aliasing between the input and the output.

void fmpz_poly_sqrlow(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to the lowest \(n\) coefficients of the square of poly.

Powering

void _fmpz_poly_pow_multinomial(fmpz *res, const fmpz *poly, slong len, ulong e)

Computes res = poly^e. This uses the J.C.P. Miller pure recurrence as follows:

If \(\ell\) is the index of the lowest non-zero coefficient in poly, as a first step this method zeros out the lowest \(e \ell\) coefficients of res. The recurrence above is then used to compute the remaining coefficients.

Assumes len > 0, e > 0. Does not support aliasing.

void fmpz_poly_pow_multinomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

Computes res = poly^e using a generalisation of binomial expansion called the J.C.P. Miller pure recurrence [1], [2]. If \(e\) is zero, returns one, so that in particular 0^0 = 1.

The formal statement of the recurrence is as follows. Write the input polynomial as \(P(x) = p_0 + p_1 x + \dotsb + p_m x^m\) with \(p_0 \neq 0\) and let

\[P(x)^n = a(n, 0) + a(n, 1) x + \dotsb + a(n, mn) x^{mn}.\]

Then \(a(n, 0) = p_0^n\) and, for all \(1 \leq k \leq mn\),

\[a(n, k) = (k p_0)^{-1} \sum_{i = 1}^m p_i \bigl( (n + 1) i - k \bigr) a(n, k-i).\]

[1] D. Knuth, The Art of Computer Programming Vol. 2, Seminumerical Algorithms, Third Edition (Reading, Massachusetts: Addison-Wesley, 1997)

[2] D. Zeilberger, The J.C.P. Miller Recurrence for Exponentiating a Polynomial, and its q-Analog, Journal of Difference Equations and Applications, 1995, Vol. 1, pp. 57–60

void _fmpz_poly_pow_binomial(fmpz *res, const fmpz *poly, ulong e)

Computes res = poly^e when poly is of length 2, using binomial expansion.

Assumes \(e > 0\). Does not support aliasing.

void fmpz_poly_pow_binomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

Computes res = poly^e when poly is of length \(2\), using binomial expansion.

If the length of poly is not \(2\), raises an exception and aborts.

void _fmpz_poly_pow_addchains(fmpz *res, const fmpz *poly, slong len, const int *a, int n)

Given a star chain \(1 = a_0 < a_1 < \dotsb < a_n = e\) computes res = poly^e.

A star chain is an addition chain \(1 = a_0 < a_1 < \dotsb < a_n\) such that, for all \(i > 0\), \(a_i = a_{i-1} + a_j\) for some \(j < i\).

Assumes that \(e > 2\), or equivalently \(n > 1\), and len > 0. Does not support aliasing.

void fmpz_poly_pow_addchains(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

Computes res = poly^e using addition chains whenever \(0 \leq e \leq 148\).

If \(e > 148\), raises an exception and aborts.

void _fmpz_poly_pow_binexp(fmpz *res, const fmpz *poly, slong len, ulong e)

Sets res = poly^e using left-to-right binary exponentiation as described on p. 461 of [Knu1997].

Assumes that len > 0, e > 1. Assumes that res is an array of length at least e*(len - 1) + 1. Does not support aliasing.

void fmpz_poly_pow_binexp(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

Computes res = poly^e using the binary exponentiation algorithm. If \(e\) is zero, returns one, so that in particular 0^0 = 1.

void _fmpz_poly_pow_small(fmpz *res, const fmpz *poly, slong len, ulong e)

Sets res = poly^e whenever \(0 \leq e \leq 4\).

Assumes that len > 0 and that res is an array of length at least e*(len - 1) + 1. Does not support aliasing.

void _fmpz_poly_pow(fmpz *res, const fmpz *poly, slong len, ulong e)

Sets res = poly^e, assuming that e, len > 0 and that res has space for e*(len - 1) + 1 coefficients. Does not support aliasing.

void fmpz_poly_pow(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

Computes res = poly^e. If \(e\) is zero, returns one, so that in particular 0^0 = 1.

void _fmpz_poly_pow_trunc(fmpz *res, const fmpz *poly, ulong e, slong n)

Sets (res, n) to (poly, n) raised to the power \(e\) and truncated to length \(n\).

Assumes that \(e, n > 0\). Allows zero-padding of (poly, n). Does not support aliasing of any inputs and outputs.

void fmpz_poly_pow_trunc(fmpz_poly_t res, const fmpz_poly_t poly, ulong e, slong n)

Notationally raises poly to the power \(e\), truncates the result to length \(n\) and writes the result in res. This is computed much more efficiently than simply powering the polynomial and truncating.

Thus, if \(n = 0\) the result is zero. Otherwise, whenever \(e = 0\) the result will be the constant polynomial equal to \(1\).

This function can be used to raise power series to a power in an efficient way.

Shifting

void _fmpz_poly_shift_left(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, len + n) to (poly, len) shifted left by \(n\) coefficients.

Inserts zero coefficients at the lower end. Assumes that len and \(n\) are positive, and that res fits len + n elements. Supports aliasing between res and poly.

void fmpz_poly_shift_left(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to poly shifted left by \(n\) coeffs. Zero coefficients are inserted.

void _fmpz_poly_shift_right(fmpz *res, const fmpz *poly, slong len, slong n)

Sets (res, len - n) to (poly, len) shifted right by \(n\) coefficients.

Assumes that len and \(n\) are positive, that len > n, and that res fits len - n elements. Supports aliasing between res and poly, although in this case the top coefficients of poly are not set to zero.

void fmpz_poly_shift_right(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Sets res to poly shifted right by \(n\) coefficients. If \(n\) is equal to or greater than the current length of poly, res is set to the zero polynomial.

Bit sizes and norms

ulong fmpz_poly_max_limbs(const fmpz_poly_t poly)

Returns the maximum number of limbs required to store the absolute value of coefficients of poly. If poly is zero, returns \(0\).

slong fmpz_poly_max_bits(const fmpz_poly_t poly)

Computes the maximum number of bits \(b\) required to store the absolute value of coefficients of poly. If all the coefficients of poly are non-negative, \(b\) is returned, otherwise \(-b\) is returned.

void fmpz_poly_height(fmpz_t height, const fmpz_poly_t poly)

Computes the height of poly, defined as the largest of the absolute values of the coefficients of poly. Equivalently, this gives the infinity norm of the coefficients. If poly is zero, the height is \(0\).

void _fmpz_poly_2norm(fmpz_t res, const fmpz *poly, slong len)

Sets res to the Euclidean norm of (poly, len), that is, the integer square root of the sum of the squares of the coefficients of poly.

void fmpz_poly_2norm(fmpz_t res, const fmpz_poly_t poly)

Sets res to the Euclidean norm of poly, that is, the integer square root of the sum of the squares of the coefficients of poly.

ulong _fmpz_poly_2norm_normalised_bits(const fmpz *poly, slong len)

Returns an upper bound on the number of bits of the normalised Euclidean norm of (poly, len), i.e. the number of bits of the Euclidean norm divided by the absolute value of the leading coefficient. The returned value will be no more than 1 bit too large.

This is used in the computation of the Landau-Mignotte bound.

It is assumed that len > 0. The result only makes sense if the leading coefficient is nonzero.

Greatest common divisor

void _fmpz_poly_gcd_subresultant(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Computes the greatest common divisor (res, len2) of (poly1, len1) and (poly2, len2), assuming len1 >= len2 > 0. The result is normalised to have positive leading coefficient. Aliasing between res, poly1 and poly2 is supported.

void fmpz_poly_gcd_subresultant(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the greatest common divisor res of poly1 and poly2, normalised to have non-negative leading coefficient.

This function uses the subresultant algorithm as described in Algorithm 3.3.1 of [Coh1996].

int _fmpz_poly_gcd_heuristic(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Computes the greatest common divisor (res, len2) of (poly1, len1) and (poly2, len2), assuming len1 >= len2 > 0. The result is normalised to have positive leading coefficient. Aliasing between res, poly1 and poly2 is not supported. The function may not always succeed in finding the GCD. If it fails, the function returns 0, otherwise it returns 1.

int fmpz_poly_gcd_heuristic(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the greatest common divisor res of poly1 and poly2, normalised to have non-negative leading coefficient.

The function may not always succeed in finding the GCD. If it fails, the function returns 0, otherwise it returns 1.

This function uses the heuristic GCD algorithm (GCDHEU). The basic strategy is to remove the content of the polynomials, pack them using Kronecker segmentation (given a bound on the size of the coefficients of the GCD) and take the integer GCD. Unpack the result and test divisibility.

void _fmpz_poly_gcd_modular(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Computes the greatest common divisor (res, len2) of (poly1, len1) and (poly2, len2), assuming len1 >= len2 > 0. The result is normalised to have positive leading coefficient. Aliasing between res, poly1 and poly2 is not supported.

void fmpz_poly_gcd_modular(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the greatest common divisor res of poly1 and poly2, normalised to have non-negative leading coefficient.

This function uses the modular GCD algorithm. The basic strategy is to remove the content of the polynomials, reduce them modulo sufficiently many primes and do CRT reconstruction until some bound is reached (or we can prove with trial division that we have the GCD).

void _fmpz_poly_gcd(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Computes the greatest common divisor res of (poly1, len1) and (poly2, len2), assuming len1 >= len2 > 0. The result is normalised to have positive leading coefficient.

Assumes that res has space for len2 coefficients. Aliasing between res, poly1 and poly2 is not supported.

void fmpz_poly_gcd(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the greatest common divisor res of poly1 and poly2, normalised to have non-negative leading coefficient.

void _fmpz_poly_xgcd_modular(fmpz_t r, fmpz *s, fmpz *t, const fmpz *f, slong len1, const fmpz *g, slong len2)

Set \(r\) to the resultant of (f, len1) and (g, len2). If the resultant is zero, the function returns immediately. Otherwise it finds polynomials \(s\) and \(t\) such that s*f + t*g = r. The length of \(s\) will be no greater than len2 and the length of \(t\) will be no greater than len1 (both are zero padded if necessary).

It is assumed that len1 >= len2 > 0. No aliasing of inputs and outputs is permitted.

The function assumes that \(f\) and \(g\) are primitive (have Gaussian content equal to 1). The result is undefined otherwise.

Uses a multimodular algorithm. The resultant is first computed and extended GCDs modulo various primes \(p\) are computed and combined using CRT. When the CRT stabilises the resulting polynomials are simply reduced modulo further primes until a proven bound is reached.

void fmpz_poly_xgcd_modular(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)

Set \(r\) to the resultant of \(f\) and \(g\). If the resultant is zero, the function then returns immediately, otherwise \(s\) and \(t\) are found such that s*f + t*g = r.

The function assumes that \(f\) and \(g\) are primitive (have Gaussian content equal to 1). The result is undefined otherwise.

Uses the multimodular algorithm.

void _fmpz_poly_xgcd(fmpz_t r, fmpz *s, fmpz *t, const fmpz *f, slong len1, const fmpz *g, slong len2)

Set \(r\) to the resultant of (f, len1) and (g, len2). If the resultant is zero, the function returns immediately. Otherwise it finds polynomials \(s\) and \(t\) such that s*f + t*g = r. The length of \(s\) will be no greater than len2 and the length of \(t\) will be no greater than len1 (both are zero padded if necessary).

The function assumes that \(f\) and \(g\) are primitive (have Gaussian content equal to 1). The result is undefined otherwise.

It is assumed that len1 >= len2 > 0. No aliasing of inputs and outputs is permitted.

void fmpz_poly_xgcd(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)

Set \(r\) to the resultant of \(f\) and \(g\). If the resultant is zero, the function then returns immediately, otherwise \(s\) and \(t\) are found such that s*f + t*g = r.

The function assumes that \(f\) and \(g\) are primitive (have Gaussian content equal to 1). The result is undefined otherwise.

void _fmpz_poly_lcm(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets (res, len1 + len2 - 1) to the least common multiple of the two polynomials (poly1, len1) and (poly2, len2), normalised to have non-negative leading coefficient.

Assumes that len1 >= len2 > 0.

Does not support aliasing.

void fmpz_poly_lcm(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the least common multiple of the two polynomials poly1 and poly2, normalised to have non-negative leading coefficient.

If either of the two polynomials is zero, sets res to zero.

This ensures that the equality

\[f g = \gcd(f, g) \operatorname{lcm}(f, g)\]

holds up to sign.

void _fmpz_poly_resultant_modular(fmpz_t res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the resultant of (poly1, len1) and (poly2, len2), assuming that len1 >= len2 > 0.

void fmpz_poly_resultant_modular(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the resultant of poly1 and poly2.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

This function uses the modular algorithm described in [Col1971].

void fmpz_poly_resultant_modular_div(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t div, slong nbits)

Computes the resultant of poly1 and poly2 divided by div using a slight modification of the above function. It is assumed that the resultant is exactly divisible by div and the result res has at most nbits bits. This bypasses the computation of general bounds.

void _fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the resultant of (poly1, len1) and (poly2, len2), assuming that len1 >= len2 > 0.

void fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the resultant of poly1 and poly2.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

This function uses the algorithm described in Algorithm 3.3.7 of [Coh1996].

void _fmpz_poly_resultant(fmpz_t res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the resultant of (poly1, len1) and (poly2, len2), assuming that len1 >= len2 > 0.

void fmpz_poly_resultant(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Computes the resultant of poly1 and poly2.

For two non-zero polynomials \(f(x) = a_m x^m + \dotsb + a_0\) and \(g(x) = b_n x^n + \dotsb + b_0\) of degrees \(m\) and \(n\), the resultant is defined to be

\[a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).\]

For convenience, we define the resultant to be equal to zero if either of the two polynomials is zero.

Discriminant

void _fmpz_poly_discriminant(fmpz_t res, const fmpz *poly, slong len)

Set res to the discriminant of (poly, len). Assumes len > 1.

void fmpz_poly_discriminant(fmpz_t res, const fmpz_poly_t poly)

Set res to the discriminant of poly. We normalise the discriminant so that \(\operatorname{disc}(f) = (-1)^{(n(n-1)/2)} \operatorname{res}(f, f')/\operatorname{lc}(f)\), thus \(\operatorname{disc}(f) = \operatorname{lc}(f)^{(2n - 2)} \prod_{i < j} (r_i - r_j)^2\), where \(\operatorname{lc}(f)\) is the leading coefficient of \(f\), \(n\) is the degree of \(f\) and \(r_i\) are the roots of \(f\).

Gaussian content

void _fmpz_poly_content(fmpz_t res, const fmpz *poly, slong len)

Sets res to the non-negative content of (poly, len). Aliasing between res and the coefficients of poly is not supported.

void fmpz_poly_content(fmpz_t res, const fmpz_poly_t poly)

Sets res to the non-negative content of poly. The content of the zero polynomial is defined to be zero. Supports aliasing, that is, res is allowed to be one of the coefficients of poly.

void _fmpz_poly_primitive_part(fmpz *res, const fmpz *poly, slong len)

Sets (res, len) to (poly, len) divided by the content of (poly, len), and normalises the result to have non-negative leading coefficient.

Assumes that (poly, len) is non-zero. Supports aliasing of res and poly.

void fmpz_poly_primitive_part(fmpz_poly_t res, const fmpz_poly_t poly)

Sets res to poly divided by the content of poly, and normalises the result to have non-negative leading coefficient. If poly is zero, sets res to zero.

Square-free

int _fmpz_poly_is_squarefree(const fmpz *poly, slong len)

Returns whether the polynomial (poly, len) is square-free.

int fmpz_poly_is_squarefree(const fmpz_poly_t poly)

Returns whether the polynomial poly is square-free. A non-zero polynomial is defined to be square-free if it has no non-unit square factors. We also define the zero polynomial to be square-free.

Returns \(1\) if the length of poly is at most \(2\). Returns whether the discriminant is zero for quadratic polynomials. Otherwise, returns whether the greatest common divisor of poly and its derivative has length \(1\).

Euclidean division

int _fmpz_poly_divrem_basecase(fmpz *Q, fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes (Q, lenA - lenB + 1), (R, lenA) such that \(A = B Q + R\) and each coefficient of \(R\) beyond lenB is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same thing as division over \(\mathbb{Q}\).

Assumes that \(\operatorname{len}(A), \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). \(R\) and \(A\) may be aliased, but apart from this no aliasing of input and output operands is allowed.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

Computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same thing as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

int _fmpz_poly_divrem_divconquer_recursive(fmpz *Q, fmpz *BQ, fmpz *W, const fmpz *A, const fmpz *B, slong lenB, int exact)

Computes (Q, lenB), (BQ, 2 lenB - 1) such that \(BQ = B \times Q\) and \(A = B Q + R\) where each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). We assume that \(\operatorname{len}(A) = 2 \operatorname{len}(B) - 1\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\).

Assumes \(\operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). Requires a temporary array (W, 2 lenB - 1). No aliasing of input and output operands is allowed.

This function does not read the bottom \(\operatorname{len}(B) - 1\) coefficients from \(A\), which means that they might not even need to exist in allocated memory.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

int _fmpz_poly_divrem_divconquer(fmpz *Q, fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes (Q, lenA - lenB + 1), (R, lenA) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\).

Assumes \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). No aliasing of input and output operands is allowed.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

Computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

int _fmpz_poly_divrem(fmpz *Q, fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes (Q, lenA - lenB + 1), (R, lenA) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same thing as division over \(\mathbb{Q}\).

Assumes \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). No aliasing of input and output operands is allowed.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_divrem(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

Computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

int _fmpz_poly_div_basecase(fmpz *Q, fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes the quotient (Q, lenA - lenB + 1) of (A, lenA) divided by (B, lenB).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\).

If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\).

Assumes \(\operatorname{len}(A), \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). Requires a temporary array \(R\) of size at least the (actual) length of \(A\). For convenience, \(R\) may be NULL. \(R\) and \(A\) may be aliased, but apart from this no aliasing of input and output operands is allowed.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_div_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

Computes the quotient \(Q\) of \(A\) divided by \(Q\).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\).

If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

int _fmpz_poly_divremlow_divconquer_recursive(fmpz *Q, fmpz *BQ, const fmpz *A, const fmpz *B, slong lenB, int exact)

Divide and conquer division of (A, 2 lenB - 1) by (B, lenB), computing only the bottom \(\operatorname{len}(B) - 1\) coefficients of \(B Q\).

Assumes \(\operatorname{len}(B) > 0\). Requires \(B Q\) to have length at least \(2 \operatorname{len}(B) - 1\), although only the bottom \(\operatorname{len}(B) - 1\) coefficients will carry meaningful output. Does not support any aliasing. Allows zero-padding in \(A\), but not in \(B\).

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

int _fmpz_poly_div_divconquer_recursive(fmpz *Q, fmpz *temp, const fmpz *A, const fmpz *B, slong lenB, int exact)

Recursive short division in the balanced case.

Computes the quotient (Q, lenB) of (A, 2 lenB - 1) upon division by (B, lenB). Requires \(\operatorname{len}(B) > 0\). Needs a temporary array temp of length \(2 \operatorname{len}(B) - 1\). Does not support any aliasing.

For further details, see [Mul2000].

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

int _fmpz_poly_div_divconquer(fmpz *Q, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes the quotient (Q, lenA - lenB + 1) of (A, lenA) upon division by (B, lenB). Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Does not support aliasing.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_div_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

Computes the quotient \(Q\) of \(A\) divided by \(B\).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\).

If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

int _fmpz_poly_div(fmpz *Q, const fmpz *A, slong lenA, const fmpz *B, slong lenB, int exact)

Computes the quotient (Q, lenA - lenB + 1) of (A, lenA) divided by (B, lenB).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\).

Assumes \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). Aliasing of input and output operands is not allowed.

If the flag exact is \(1\), the function stops if an inexact division is encountered, upon which the function will return \(0\). If no inexact division is encountered, the function returns \(1\). Note that this does not guarantee the remainder of the polynomial division is zero, merely that its length is less than that of B. This feature is useful for series division and for divisibility testing (upon testing the remainder).

For ordinary use set the flag exact to \(0\). In this case, no checks or early aborts occur and the function always returns \(1\).

void fmpz_poly_div(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

Computes the quotient \(Q\) of \(A\) divided by \(B\).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(Q\). An exception is raised if \(B\) is zero.

void _fmpz_poly_rem_basecase(fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

Computes the remainder (R, lenA) of (A, lenA) upon division by (B, lenB).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same thing as division over \(\mathbb{Q}\).

Assumes that \(\operatorname{len}(A), \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). \(R\) and \(A\) may be aliased, but apart from this no aliasing of input and output operands is allowed.

void fmpz_poly_rem_basecase(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

Computes the remainder \(R\) of \(A\) upon division by \(B\).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

void _fmpz_poly_rem(fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

Computes the remainder (R, lenA) of (A, lenA) upon division by (B, lenB).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same thing as division over \(\mathbb{Q}\).

Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). Aliasing of input and output operands is not allowed.

void fmpz_poly_rem(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

Computes the remainder \(R\) of \(A\) upon division by \(B\).

Notationally, computes \(Q\), \(R\) such that \(A = B Q + R\) and each coefficient of \(R\) beyond \(\operatorname{len}(B) - 1\) is reduced modulo the leading coefficient of \(B\). If the leading coefficient of \(B\) is \(\pm 1\) or the division is exact, this is the same as division over \(\mathbb{Q}\). An exception is raised if \(B\) is zero.

void _fmpz_poly_div_root(fmpz *Q, const fmpz *A, slong len, const fmpz_t c)

Computes the quotient (Q, len-1) of (A, len) upon division by \(x - c\).

Supports aliasing of Q and A, but the result is undefined in case of partial overlap.

void fmpz_poly_div_root(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_t c)

Computes the quotient (Q, len-1) of (A, len) upon division by \(x - c\).

void _fmpz_poly_divexact(fmpz *Q, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

void fmpz_poly_divexact(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

Like fmpz_poly_div(), but assumes that the division is exact.

Division with precomputed inverse

void _fmpz_poly_preinvert(fmpz *B_inv, const fmpz *B, slong n)

Given a monic polynomial B of length n, compute a precomputed inverse B_inv of length n for use in the functions below. No aliasing of B and B_inv is permitted. We assume n is not zero.

void fmpz_poly_preinvert(fmpz_poly_t B_inv, const fmpz_poly_t B)

Given a monic polynomial B, compute a precomputed inverse B_inv for use in the functions below. An exception is raised if B is zero.

void _fmpz_poly_div_preinv(fmpz *Q, const fmpz *A, slong len1, const fmpz *B, const fmpz *B_inv, slong len2)

Given a precomputed inverse B_inv of the polynomial B of length len2, compute the quotient Q of A by B. We assume the length len1 of A is at least len2. The polynomial Q must have space for len1 - len2 + 1 coefficients. No aliasing of operands is permitted.

void fmpz_poly_div_preinv(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)

Given a precomputed inverse B_inv of the polynomial B, compute the quotient Q of A by B. Aliasing of B and B_inv is not permitted.

void _fmpz_poly_divrem_preinv(fmpz *Q, fmpz *A, slong len1, const fmpz *B, const fmpz *B_inv, slong len2)

Given a precomputed inverse B_inv of the polynomial B of length len2, compute the quotient Q of A by B. The remainder is then placed in A. We assume the length len1 of A is at least len2. The polynomial Q must have space for len1 - len2 + 1 coefficients. No aliasing of operands is permitted.

void fmpz_poly_divrem_preinv(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)

Given a precomputed inverse B_inv of the polynomial B, compute the quotient Q of A by B and the remainder R. Aliasing of B and B_inv is not permitted.

fmpz **_fmpz_poly_powers_precompute(const fmpz *B, slong len)

Computes 2*len - 1 powers of \(x\) modulo the polynomial \(B\) of the given length. This is used as a kind of precomputed inverse in the remainder routine below.

void fmpz_poly_powers_precompute(fmpz_poly_powers_precomp_t pinv, fmpz_poly_t poly)

Computes 2*len - 1 powers of \(x\) modulo the polynomial \(B\) of the given length. This is used as a kind of precomputed inverse in the remainder routine below.

void _fmpz_poly_powers_clear(fmpz **powers, slong len)

Clean up resources used by precomputed powers which have been computed by _fmpz_poly_powers_precompute.

void fmpz_poly_powers_clear(fmpz_poly_powers_precomp_t pinv)

Clean up resources used by precomputed powers which have been computed by fmpz_poly_powers_precompute.

void _fmpz_poly_rem_powers_precomp(fmpz *A, slong m, const fmpz *B, slong n, fmpz **const powers)

Set \(A\) to the remainder of \(A\) divide \(B\) given precomputed powers mod \(B\) provided by _fmpz_poly_powers_precompute. No aliasing is allowed.

void fmpz_poly_rem_powers_precomp(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_powers_precomp_t B_inv)

Set \(R\) to the remainder of \(A\) divide \(B\) given precomputed powers mod \(B\) provided by fmpz_poly_powers_precompute.

Divisibility testing

int _fmpz_poly_divides(fmpz *Q, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

Returns 1 if (B, lenB) divides (A, lenA) exactly and sets \(Q\) to the quotient, otherwise returns 0.

It is assumed that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\) and that \(Q\) has space for \(\operatorname{len}(A) - \operatorname{len}(B) + 1\) coefficients.

Aliasing of \(Q\) with either of the inputs is not permitted.

This function is currently unoptimised and provided for convenience only.

int fmpz_poly_divides(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

Returns 1 if \(B\) divides \(A\) exactly and sets \(Q\) to the quotient, otherwise returns 0.

This function is currently unoptimised and provided for convenience only.

slong fmpz_poly_remove(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Set res to poly1 divided by the highest power of poly2 that divides it and return the power. The divisor poly2 must not be zero or \(\pm 1\), otherwise an exception is raised.

Division mod p

void fmpz_poly_divlow_smodp(fmpz *res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)

Compute the \(n\) lowest coefficients of \(f\) divided by \(g\), assuming the division is exact modulo \(p\). The computed coefficients are reduced modulo \(p\) using the symmetric remainder system. We require \(f\) to be at least \(n\) in length. The function can handle trailing zeroes, but the low nonzero coefficient of \(g\) must be coprime to \(p\). This is a bespoke function used by factoring.

void fmpz_poly_divhigh_smodp(fmpz *res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)

Compute the \(n\) highest coefficients of \(f\) divided by \(g\), assuming the division is exact modulo \(p\). The computed coefficients are reduced modulo \(p\) using the symmetric remainder system. We require \(f\) to be as output by fmpz_poly_mulhigh_n given polynomials \(g\) and a polynomial of length \(n\) as inputs. The leading coefficient of \(g\) must be coprime to \(p\). This is a bespoke function used by factoring.

Power series division

void _fmpz_poly_inv_series_basecase(fmpz *Qinv, const fmpz *Q, slong Qlen, slong n)

Computes the first \(n\) terms of the inverse power series of (Q, lenQ) using a recurrence.

Assumes that \(n \geq 1\) and that \(Q\) has constant term \(\pm 1\). Does not support aliasing.

void fmpz_poly_inv_series_basecase(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

Computes the first \(n\) terms of the inverse power series of \(Q\) using a recurrence, assuming that \(Q\) has constant term \(\pm 1\) and \(n \geq 1\).

void _fmpz_poly_inv_series_newton(fmpz *Qinv, const fmpz *Q, slong Qlen, slong n)

Computes the first \(n\) terms of the inverse power series of (Q, lenQ) using Newton iteration.

Assumes that \(n \geq 1\) and that \(Q\) has constant term \(\pm 1\). Does not support aliasing.

void fmpz_poly_inv_series_newton(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

Computes the first \(n\) terms of the inverse power series of \(Q\) using Newton iteration, assuming \(Q\) has constant term \(\pm 1\) and \(n \geq 1\).

void _fmpz_poly_inv_series(fmpz *Qinv, const fmpz *Q, slong Qlen, slong n)

Computes the first \(n\) terms of the inverse power series of (Q, lenQ).

Assumes that \(n \geq 1\) and that \(Q\) has constant term \(\pm 1\). Does not support aliasing.

void fmpz_poly_inv_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

Computes the first \(n\) terms of the inverse power series of \(Q\), assuming \(Q\) has constant term \(\pm 1\) and \(n \geq 1\).

void _fmpz_poly_div_series_basecase(fmpz *Q, const fmpz *A, slong Alen, const fmpz *B, slong Blen, slong n)

void _fmpz_poly_div_series_divconquer(fmpz *Q, const fmpz *A, slong Alen, const fmpz *B, slong Blen, slong n)

void _fmpz_poly_div_series(fmpz *Q, const fmpz *A, slong Alen, const fmpz *B, slong Blen, slong n)

Divides (A, Alen) by (B, Blen) as power series over \(\mathbb{Z}\), assuming \(B\) has constant term \(\pm 1\) and \(n \geq 1\). Aliasing is not supported.

void fmpz_poly_div_series_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

void fmpz_poly_div_series_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

void fmpz_poly_div_series(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

Performs power series division in \(\mathbb{Z}[[x]] / (x^n)\). The function considers the polynomials \(A\) and \(B\) as power series of length \(n\) starting with the constant terms. The function assumes that \(B\) has constant term \(\pm 1\) and \(n \geq 1\).

Pseudo division

void _fmpz_poly_pseudo_divrem_basecase(fmpz *Q, fmpz *R, ulong *d, const fmpz *A, slong lenA, const fmpz *B, slong lenB, const fmpz_preinvn_t inv)

If \(\ell\) is the leading coefficient of \(B\), then computes \(Q\), \(R\) such that \(\ell^d A = Q B + R\). This function is used for simulating division over \(\mathbb{Q}\).

Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Assumes that \(Q\) can fit \(\operatorname{len}(A) - \operatorname{len}(B) + 1\) coefficients, and that \(R\) can fit \(\operatorname{len}(A)\) coefficients. Supports aliasing of (R, lenA) and (A, lenA). But other than this, no aliasing of the inputs and outputs is supported.

An optional precomputed inverse of the leading coefficient of \(B\) from fmpz_preinvn_init can be supplied. Otherwise inv should be NULL.

Note: fmpz.h has to be included before fmpz_poly.h in order for fmpz_poly.h to declare this function.

void fmpz_poly_pseudo_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, ulong *d, const fmpz_poly_t A, const fmpz_poly_t B)

If \(\ell\) is the leading coefficient of \(B\), then computes \(Q\), \(R\) such that \(\ell^d A = Q B + R\). This function is used for simulating division over \(\mathbb{Q}\).

void _fmpz_poly_pseudo_divrem_divconquer(fmpz *Q, fmpz *R, ulong *d, const fmpz *A, slong lenA, const fmpz *B, slong lenB, const fmpz_preinvn_t inv)

Computes (Q, lenA - lenB + 1), (R, lenA) such that \(\ell^d A = B Q + R\), only setting the bottom \(\operatorname{len}(B) - 1\) coefficients of \(R\) to their correct values. The remaining top coefficients of (R, lenA) may be arbitrary.

Assumes \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Allows zero-padding in (A, lenA). No aliasing of input and output operands is allowed.

An optional precomputed inverse of the leading coefficient of \(B\) from fmpz_preinvn_init can be supplied. Otherwise inv should be NULL.

Note: fmpz.h has to be included before fmpz_poly.h in order for fmpz_poly.h to declare this function.

void fmpz_poly_pseudo_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, ulong *d, const fmpz_poly_t A, const fmpz_poly_t B)

Computes \(Q\), \(R\), and \(d\) such that \(\ell^d A = B Q + R\), where \(R\) has length less than the length of \(B\) and \(\ell\) is the leading coefficient of \(B\). An exception is raised if \(B\) is zero.

void _fmpz_poly_pseudo_divrem_cohen(fmpz *Q, fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Assumes that \(Q\) can fit \(\operatorname{len}(A) - \operatorname{len}(B) + 1\) coefficients, and that \(R\) can fit \(\operatorname{len}(A)\) coefficients. Supports aliasing of (R, lenA) and (A, lenA). But other than this, no aliasing of the inputs and outputs is supported.

void fmpz_poly_pseudo_divrem_cohen(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

This is a variant of fmpz_poly_pseudo_divrem which computes polynomials \(Q\) and \(R\) such that \(\ell^d A = B Q + R\). However, the value of \(d\) is fixed at \(\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}\).

This function is faster when the remainder is not well behaved, i.e. where it is not expected to be close to zero. Note that this function is not asymptotically fast. It is efficient only for short polynomials, e.g. when \(\operatorname{len}(B) < 32\).

void _fmpz_poly_pseudo_rem_cohen(fmpz *R, const fmpz *A, slong lenA, const fmpz *B, slong lenB)

Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Assumes that \(R\) can fit \(\operatorname{len}(A)\) coefficients. Supports aliasing of (R, lenA) and (A, lenA). But other than this, no aliasing of the inputs and outputs is supported.

void fmpz_poly_pseudo_rem_cohen(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

This is a variant of fmpz_poly_pseudo_rem() which computes polynomials \(Q\) and \(R\) such that \(\ell^d A = B Q + R\), but only returns \(R\). However, the value of \(d\) is fixed at \(\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}\).

This function is faster when the remainder is not well behaved, i.e. where it is not expected to be close to zero. Note that this function is not asymptotically fast. It is efficient only for short polynomials, e.g. when \(\operatorname{len}(B) < 32\).

This function uses the algorithm described in Algorithm 3.1.2 of [Coh1996].

void _fmpz_poly_pseudo_divrem(fmpz *Q, fmpz *R, ulong *d, const fmpz *A, slong lenA, const fmpz *B, slong lenB, const fmpz_preinvn_t inv)

If \(\ell\) is the leading coefficient of \(B\), then computes (Q, lenA - lenB + 1), (R, lenB - 1) and \(d\) such that \(\ell^d A = B Q + R\). This function is used for simulating division over \(\mathbb{Q}\).

Assumes that \(\operatorname{len}(A) \geq \operatorname{len}(B) > 0\). Assumes that \(Q\) can fit \(\operatorname{len}(A) - \operatorname{len}(B) + 1\) coefficients, and that \(R\) can fit \(\operatorname{len}(A)\) coefficients, although on exit only the bottom \(\operatorname{len}(B)\) coefficients will carry meaningful data.

Supports aliasing of (R, lenA) and (A, lenA). But other than this, no aliasing of the inputs and outputs is supported.

An optional precomputed inverse of the leading coefficient of \(B\) from fmpz_preinvn_init can be supplied. Otherwise inv should be NULL.

Note: fmpz.h has to be included before fmpz_poly.h in order for fmpz_poly.h to declare this function.

void fmpz_poly_pseudo_divrem(fmpz_poly_t Q, fmpz_poly_t R, ulong *d, const fmpz_poly_t A, const fmpz_poly_t B)

Computes \(Q\), \(R\), and \(d\) such that \(\ell^d A = B Q + R\).

void _fmpz_poly_pseudo_div(fmpz *Q, ulong *d, const fmpz *A, slong lenA, const fmpz *B, slong lenB, const fmpz_preinvn_t inv)

Pseudo-division, only returning the quotient.

Note: fmpz.h has to be included before fmpz_poly.h in order for fmpz_poly.h to declare this function.

void fmpz_poly_pseudo_div(fmpz_poly_t Q, ulong *d, const fmpz_poly_t A, const fmpz_poly_t B)

Pseudo-division, only returning the quotient.

void _fmpz_poly_pseudo_rem(fmpz *R, ulong *d, const fmpz *A, slong lenA, const fmpz *B, slong lenB, const fmpz_preinvn_t inv)

Pseudo-division, only returning the remainder.

Note: fmpz.h has to be included before fmpz_poly.h in order for fmpz_poly.h to declare this function.

void fmpz_poly_pseudo_rem(fmpz_poly_t R, ulong *d, const fmpz_poly_t A, const fmpz_poly_t B)

Pseudo-division, only returning the remainder.

Derivative

void _fmpz_poly_derivative(fmpz *rpoly, const fmpz *poly, slong len)

Sets (rpoly, len - 1) to the derivative of (poly, len). Also handles the cases where len is \(0\) or \(1\) correctly. Supports aliasing of rpoly and poly.

void fmpz_poly_derivative(fmpz_poly_t res, const fmpz_poly_t poly)

Sets res to the derivative of poly.

void _fmpz_poly_nth_derivative(fmpz *rpoly, const fmpz *poly, ulong n, slong len)

Sets (rpoly, len - n) to the nth derivative of (poly, len). Also handles the cases where len <= n correctly. Supports aliasing of rpoly and poly.

void fmpz_poly_nth_derivative(fmpz_poly_t res, const fmpz_poly_t poly, ulong n)

Sets res to the nth derivative of poly.

Evaluation

void _fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz *poly, slong len, const fmpz_t a)

Evaluates the polynomial (poly, len) at the integer \(a\) using a divide and conquer approach. Assumes that the length of the polynomial is at least one. Allows zero padding. Does not allow aliasing between res and x.

void fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz_poly_t poly, const fmpz_t a)

Evaluates the polynomial poly at the integer \(a\) using a divide and conquer approach.

Aliasing between res and a is supported, however, res may not be part of poly.

void _fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz *f, slong len, const fmpz_t a)

Evaluates the polynomial (f, len) at the integer \(a\) using Horner’s rule, and sets res to the result. Aliasing between res and \(a\) or any of the coefficients of \(f\) is not supported.

void fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)

Evaluates the polynomial \(f\) at the integer \(a\) using Horner’s rule, and sets res to the result.

As expected, aliasing between res and a is supported. However, res may not be aliased with a coefficient of \(f\).

void _fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz *f, slong len, const fmpz_t a)

Evaluates the polynomial (f, len) at the integer \(a\) and sets res to the result. Aliasing between res and \(a\) or any of the coefficients of \(f\) is not supported.

void fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)

Evaluates the polynomial \(f\) at the integer \(a\) and sets res to the result.

As expected, aliasing between res and \(a\) is supported. However, res may not be aliased with a coefficient of \(f\).

void _fmpz_poly_evaluate_divconquer_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz *f, slong len, const fmpz_t anum, const fmpz_t aden)

Evaluates the polynomial (f, len) at the rational (anum, aden) using a divide and conquer approach, and sets (rnum, rden) to the result in lowest terms. Assumes that the length of the polynomial is at least one.

Aliasing between (rnum, rden) and (anum, aden) or any of the coefficients of \(f\) is not supported.

void fmpz_poly_evaluate_divconquer_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

Evaluates the polynomial \(f\) at the rational \(a\) using a divide and conquer approach, and sets res to the result.

void _fmpz_poly_evaluate_horner_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz *f, slong len, const fmpz_t anum, const fmpz_t aden)

Evaluates the polynomial (f, len) at the rational (anum, aden) using Horner’s rule, and sets (rnum, rden) to the result in lowest terms.

Aliasing between (rnum, rden) and (anum, aden) or any of the coefficients of \(f\) is not supported.

void fmpz_poly_evaluate_horner_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

Evaluates the polynomial \(f\) at the rational \(a\) using Horner’s rule, and sets res to the result.

void _fmpz_poly_evaluate_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz *f, slong len, const fmpz_t anum, const fmpz_t aden)

Evaluates the polynomial (f, len) at the rational (anum, aden) and sets (rnum, rden) to the result in lowest terms.

Aliasing between (rnum, rden) and (anum, aden) or any of the coefficients of \(f\) is not supported.

void fmpz_poly_evaluate_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

Evaluates the polynomial \(f\) at the rational \(a\), and sets res to the result.

ulong _fmpz_poly_evaluate_mod(const fmpz *poly, slong len, ulong a, ulong n, ulong ninv)

Evaluates (poly, len) at the value \(a\) modulo \(n\) and returns the result. The last argument ninv must be set to the precomputed inverse of \(n\), which can be obtained using the function n_preinvert_limb().

ulong fmpz_poly_evaluate_mod(const fmpz_poly_t poly, ulong a, ulong n)

Evaluates poly at the value \(a\) modulo \(n\) and returns the result.

void fmpz_poly_evaluate_fmpz_vec(fmpz *res, const fmpz_poly_t f, const fmpz *a, slong n)

Evaluates f at the \(n\) values given in the vector f, writing the results to res.

double _fmpz_poly_evaluate_horner_d(const fmpz *poly, slong n, double d)

Evaluate (poly, n) at the double \(d\). No attempt is made to do this efficiently or in a numerically stable way. It is currently only used in Flint for quick and dirty evaluations of polynomials with all coefficients positive.

double fmpz_poly_evaluate_horner_d(const fmpz_poly_t poly, double d)

Evaluate poly at the double \(d\). No attempt is made to do this efficiently or in a numerically stable way. It is currently only used in Flint for quick and dirty evaluations of polynomials with all coefficients positive.

double _fmpz_poly_evaluate_horner_d_2exp(slong *exp, const fmpz *poly, slong n, double d)

Evaluate (poly, n) at the double \(d\). Return the result as a double and an exponent exp combination. No attempt is made to do this efficiently or in a numerically stable way. It is currently only used in Flint for quick and dirty evaluations of polynomials with all coefficients positive.

double fmpz_poly_evaluate_horner_d_2exp(slong *exp, const fmpz_poly_t poly, double d)

Evaluate poly at the double \(d\). Return the result as a double and an exponent exp combination. No attempt is made to do this efficiently or in a numerically stable way. It is currently only used in Flint for quick and dirty evaluations of polynomials with all coefficients positive.

double _fmpz_poly_evaluate_horner_d_2exp2(slong *exp, const fmpz *poly, slong n, double d, slong dexp)

Evaluate poly at d*2^dexp. Return the result as a double and an exponent exp combination. No attempt is made to do this efficiently or in a numerically stable way. It is currently only used in Flint for quick and dirty evaluations of polynomials with all coefficients positive.

Newton basis

void _fmpz_poly_monomial_to_newton(fmpz *poly, const fmpz *roots, slong n)

Converts (poly, n) in-place from its coefficients given in the standard monomial basis to the Newton basis for the roots \(r_0, r_1, \ldots, r_{n-2}\). In other words, this determines output coefficients \(c_i\) such that \(c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})\) is equal to the input polynomial. Uses repeated polynomial division.

void _fmpz_poly_newton_to_monomial(fmpz *poly, const fmpz *roots, slong n)

Converts (poly, n) in-place from its coefficients given in the Newton basis for the roots \(r_0, r_1, \ldots, r_{n-2}\) to the standard monomial basis. In other words, this evaluates \(c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})\) where \(c_i\) are the input coefficients for poly. Uses Horner’s rule.

Interpolation

void fmpz_poly_interpolate_fmpz_vec(fmpz_poly_t poly, const fmpz *xs, const fmpz *ys, slong n)

Sets poly to the unique interpolating polynomial of degree at most \(n - 1\) satisfying \(f(x_i) = y_i\) for every pair \(x_i, y_u\) in xs and ys, assuming that this polynomial has integer coefficients.

If an interpolating polynomial with integer coefficients does not exist, a FLINT_INEXACT exception is thrown.

It is assumed that the \(x\) values are distinct.

Composition

void _fmpz_poly_compose_horner(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the composition of (poly1, len1) and (poly2, len2).

Assumes that res has space for (len1-1)*(len2-1) + 1 coefficients. Assumes that poly1 and poly2 are non-zero polynomials. Does not support aliasing between any of the inputs and the output.

void fmpz_poly_compose_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the composition of poly1 and poly2. To be more precise, denoting res, poly1, and poly2 by \(f\), \(g\), and \(h\), sets \(f(t) = g(h(t))\).

This implementation uses Horner’s method.

void _fmpz_poly_compose_divconquer(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Computes the composition of (poly1, len1) and (poly2, len2) using a divide and conquer approach and places the result into res, assuming res can hold the output of length (len1 - 1) * (len2 - 1) + 1.

Assumes len1, len2 > 0. Does not support aliasing between res and any of (poly1, len1) and (poly2, len2).

void fmpz_poly_compose_divconquer(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the composition of poly1 and poly2. To be precise about the order of composition, denoting res, poly1, and poly2 by \(f\), \(g\), and \(h\), respectively, sets \(f(t) = g(h(t))\).

void _fmpz_poly_compose(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2)

Sets res to the composition of (poly1, len1) and (poly2, len2).

Assumes that res has space for (len1-1)*(len2-1) + 1 coefficients. Assumes that poly1 and poly2 are non-zero polynomials. Does not support aliasing between any of the inputs and the output.

void fmpz_poly_compose(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

Sets res to the composition of poly1 and poly2. To be precise about the order of composition, denoting res, poly1, and poly2 by \(f\), \(g\), and \(h\), respectively, sets \(f(t) = g(h(t))\).

Inflation and deflation

void fmpz_poly_inflate(fmpz_poly_t result, const fmpz_poly_t input, ulong inflation)

Sets result to the inflated polynomial \(p(x^n)\) where \(p\) is given by input and \(n\) is given by inflation.

void fmpz_poly_deflate(fmpz_poly_t result, const fmpz_poly_t input, ulong deflation)

Sets result to the deflated polynomial \(p(x^{1/n})\) where \(p\) is given by input and \(n\) is given by deflation. Requires \(n > 0\).

ulong fmpz_poly_deflation(const fmpz_poly_t input)

Returns the largest integer by which input can be deflated. As special cases, returns 0 if input is the zero polynomial and 1 if input is a constant polynomial.

Taylor shift

void _fmpz_poly_taylor_shift_horner(fmpz *poly, const fmpz_t c, slong n)

Performs the Taylor shift composing poly by \(x+c\) in-place. Uses an efficient version Horner’s rule.

void fmpz_poly_taylor_shift_horner(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

Performs the Taylor shift composing f by \(x+c\).

void _fmpz_poly_taylor_shift_divconquer(fmpz *poly, const fmpz_t c, slong n)

Performs the Taylor shift composing poly by \(x+c\) in-place. Uses the divide-and-conquer polynomial composition algorithm.

void fmpz_poly_taylor_shift_divconquer(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

Performs the Taylor shift composing f by \(x+c\). Uses the divide-and-conquer polynomial composition algorithm.

void _fmpz_poly_taylor_shift_multi_mod(fmpz *poly, const fmpz_t c, slong n)

Performs the Taylor shift composing poly by \(x+c\) in-place. Uses a multimodular algorithm, distributing the computation across flint_get_num_threads() threads.

void fmpz_poly_taylor_shift_multi_mod(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

Performs the Taylor shift composing f by \(x+c\). Uses a multimodular algorithm, distributing the computation across flint_get_num_threads() threads.

void _fmpz_poly_taylor_shift(fmpz *poly, const fmpz_t c, slong n)

Performs the Taylor shift composing poly by \(x+c\) in-place.

void fmpz_poly_taylor_shift(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

Performs the Taylor shift composing f by \(x+c\).

Power series composition

void _fmpz_poly_compose_series_horner(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation uses the Horner scheme.

void fmpz_poly_compose_series_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

This implementation uses the Horner scheme.

void _fmpz_poly_compose_series_brent_kung(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978].

void fmpz_poly_compose_series_brent_kung(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978].

void _fmpz_poly_compose_series(fmpz *res, const fmpz *poly1, slong len1, const fmpz *poly2, slong len2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

Assumes that len1, len2, n > 0, that len1, len2 <= n, and that (len1-1) * (len2-1) + 1 <= n, and that res has space for n coefficients. Does not support aliasing between any of the inputs and the output.

This implementation automatically switches between the Horner scheme and Brent-Kung algorithm 2.1 depending on the size of the inputs.

void fmpz_poly_compose_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

Sets res to the composition of poly1 and poly2 modulo \(x^n\), where the constant term of poly2 is required to be zero.

This implementation automatically switches between the Horner scheme and Brent-Kung algorithm 2.1 depending on the size of the inputs.

Power series reversion

void _fmpz_poly_revert_series(fmpz *Qinv, const fmpz *Q, slong Qlen, slong n)

void fmpz_poly_revert_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

Sets Qinv to the compositional inverse or reversion of Q as a power series, i.e. computes \(Q^{-1}\) such that \(Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n\). It is required that \(Q_0 = 0\) and \(Q_1 = \pm 1\).

Wraps _gr_poly_revert_series() which chooses automatically between various algorithms.

Square root

int _fmpz_poly_sqrtrem_classical(fmpz *res, fmpz *r, const fmpz *poly, slong len)

Returns 1 if (poly, len) can be written in the form \(A^2 + R\) where deg(\(R\)) < deg(poly), otherwise returns \(0\). If it can be so written, (res, m - 1) is set to \(A\) and (res, m) is set to \(R\), where \(m = \deg(\mathtt{poly})/2 + 1\).

For efficiency reasons, r must have room for len coefficients, and may alias poly.

int fmpz_poly_sqrtrem_classical(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)

If \(a\) can be written as \(b^2 + r\) with \(\deg(r) < \deg(a)/2\), return \(1\) and set \(b\) and \(r\) appropriately. Otherwise return \(0\).

int _fmpz_poly_sqrtrem_divconquer(fmpz *res, fmpz *r, const fmpz *poly, slong len, fmpz *temp)

Returns 1 if (poly, len) can be written in the form \(A^2 + R\) where deg(\(R\)) < deg(poly), otherwise returns \(0\). If it can be so written, (res, m - 1) is set to \(A\) and (res, m) is set to \(R\), where \(m = \deg(\mathtt{poly})/2 + 1\).

For efficiency reasons, r must have room for len coefficients, and may alias poly. Temporary space of len coefficients is required.

int fmpz_poly_sqrtrem_divconquer(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)

If \(a\) can be written as \(b^2 + r\) with \(\deg(r) < \deg(a)/2\), return \(1\) and set \(b\) and \(r\) appropriately. Otherwise return \(0\).

int _fmpz_poly_sqrt_classical(fmpz *res, const fmpz *poly, slong len, int exact)

If exact is \(1\) and (poly, len) is a perfect square, sets (res, len / 2 + 1) to the square root of poly with positive leading coefficient and returns 1. Otherwise returns 0.

If exact is \(0\), allows a remainder after the square root, which is not computed.

This function first uses various tests to detect nonsquares quickly. Then, it computes the square root iteratively from top to bottom, requiring \(O(n^2)\) coefficient operations.

int fmpz_poly_sqrt_classical(fmpz_poly_t b, const fmpz_poly_t a)

If a is a perfect square, sets b to the square root of a with positive leading coefficient and returns 1. Otherwise returns 0.

int _fmpz_poly_sqrt_KS(fmpz *res, const fmpz *poly, slong len)

Heuristic square root. If the return value is \(-1\), the function failed, otherwise it succeeded and the following applies.

If (poly, len) is a perfect square, sets (res, len / 2 + 1) to the square root of poly with positive leading coefficient and returns 1. Otherwise returns 0.

This function first uses various tests to detect nonsquares quickly. Then, it computes the square root iteratively from top to bottom.

int fmpz_poly_sqrt_KS(fmpz_poly_t b, const fmpz_poly_t a)

Heuristic square root. If the return value is \(-1\), the function failed, otherwise it succeeded and the following applies.

If a is a perfect square, sets b to the square root of a with positive leading coefficient and returns 1. Otherwise returns 0.

int _fmpz_poly_sqrt_divconquer(fmpz *res, const fmpz *poly, slong len, int exact)

If exact is \(1\) and (poly, len) is a perfect square, sets (res, len / 2 + 1) to the square root of poly with positive leading coefficient and returns 1. Otherwise returns 0.

If exact is \(0\), allows a remainder after the square root, which is not computed.

This function first uses various tests to detect nonsquares quickly. Then, it computes the square root iteratively from top to bottom.

int fmpz_poly_sqrt_divconquer(fmpz_poly_t b, const fmpz_poly_t a)

If a is a perfect square, sets b to the square root of a with positive leading coefficient and returns 1. Otherwise returns 0.

int _fmpz_poly_sqrt(fmpz *res, const fmpz *poly, slong len)

If (poly, len) is a perfect square, sets (res, len / 2 + 1) to the square root of poly with positive leading coefficient and returns 1. Otherwise returns 0.

int fmpz_poly_sqrt(fmpz_poly_t b, const fmpz_poly_t a)

If a is a perfect square, sets b to the square root of a with positive leading coefficient and returns 1. Otherwise returns 0.

int _fmpz_poly_sqrt_series(fmpz *res, const fmpz *poly, slong len, slong n)

Set (res, n) to the square root of the series (poly, n), if it exists, and return \(1\), otherwise, return \(0\).

If the valuation of poly is not zero, res is zero padded to make up for the fact that the square root may not be known to precision \(n\).

int fmpz_poly_sqrt_series(fmpz_poly_t b, const fmpz_poly_t a, slong n)

Set b to the square root of the series a, where the latter is taken to be a series of precision \(n\). If such a square root exists, return \(1\), otherwise, return \(0\).

Note that if the valuation of a is not zero, b will not have precision n. It is given only to the precision to which the square root can be computed.

Power sums

void _fmpz_poly_power_sums_naive(fmpz *res, const fmpz *poly, slong len, slong n)

Compute the (truncated) power sums series of the monic polynomial (poly,len) up to length \(n\) using Newton identities.

void fmpz_poly_power_sums_naive(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Compute the (truncated) power sum series of the monic polynomial poly up to length \(n\) using Newton identities.

void fmpz_poly_power_sums(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

Compute the (truncated) power sums series of the monic polynomial poly up to length \(n\). That is the power series whose coefficient of degree \(i\) is the sum of the \(i\)-th power of all (complex) roots of the polynomial poly.

void _fmpz_poly_power_sums_to_poly(fmpz *res, const fmpz *poly, slong len)

Compute the (monic) polynomial given by its power sums series (poly,len).

void fmpz_poly_power_sums_to_poly(fmpz_poly_t res, const fmpz_poly_t Q)

Compute the (monic) polynomial given its power sums series (Q).

Signature

void _fmpz_poly_signature(slong *r1, slong *r2, const fmpz *poly, slong len)

Computes the signature \((r_1, r_2)\) of the polynomial (poly, len). Assumes that the polynomial is squarefree over \(\mathbb{Q}\).

void fmpz_poly_signature(slong *r1, slong *r2, const fmpz_poly_t poly)

Computes the signature \((r_1, r_2)\) of the polynomial poly, which is assumed to be square-free over \(\mathbb{Q}\). The values of \(r_1\) and \(2 r_2\) are the number of real and complex roots of the polynomial, respectively. For convenience, the zero polynomial is allowed, in which case the output is \((0, 0)\).

If the polynomial is not square-free, the behaviour is undefined and an exception may be raised.

This function uses the algorithm described in Algorithm 4.1.11 of [Coh1996].

Hensel lifting

void fmpz_poly_hensel_build_tree(slong *link, fmpz_poly_t *v, fmpz_poly_t *w, const nmod_poly_factor_t fac)

Initialises and builds a Hensel tree consisting of two arrays \(v\), \(w\) of polynomials and an array of links, called link.

The caller supplies a set of \(r\) local factors (in the factor structure fac) of some polynomial \(F\) over \(\mathbf{Z}\). They also supply two arrays of initialised polynomials \(v\) and \(w\), each of length \(2r - 2\) and an array link, also of length \(2r - 2\).

We will have five arrays: a \(v\) of fmpz_poly_t’s and a \(V\) of nmod_poly_t’s and also a \(w\) and a \(W\) and link. Here’s the idea: we sort each leaf and node of a factor tree by degree, in fact choosing to multiply the two smallest factors, then the next two smallest (factors or products) etc. until a tree is made. The tree will be stored in the \(v\)’s. The first two elements of \(v\) will be the smallest modular factors, the last two elements of \(v\) will multiply to form \(F\) itself. Since \(v\) will be rearranging the original factors we will need to be able to recover the original order. For this we use the array link which has nonnegative even numbers and negative numbers. It is an array of slongs which aligns with \(V\) and \(v\) if link has a negative number in spot \(j\) that means \(V_j\) is an original modular factor which has been lifted, if link[j] is a nonnegative even number then \(V_j\) stores a product of the two entries at V[link[j]] and V[link[j]+1]. \(W\) and \(w\) play the role of the extended GCD, at \(V_0\), \(V_2\), \(V_4\), etc. we have a new product, \(W_0\), \(W_2\), \(W_4\), etc. are the XGCD cofactors of the \(V\)’s. For example, \(V_0 W_0 + V_1 W_1 \equiv 1 \pmod{p^{\ell}}\) for some \(\ell\). These will be lifted along with the entries in \(V\). It is not enough to just lift each factor, we have to lift the entire tree and the tree of XGCD cofactors.

void fmpz_poly_hensel_lift(fmpz_poly_t G, fmpz_poly_t H, fmpz_poly_t A, fmpz_poly_t B, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

This is the main Hensel lifting routine, which performs a Hensel step from polynomials mod \(p\) to polynomials mod \(P = p p_1\). One starts with polynomials \(f\), \(g\), \(h\) such that \(f = gh \pmod p\). The polynomials \(a\), \(b\) satisfy \(ag + bh = 1 \pmod p\).

The lifting formulae are

\[ \begin{align}\begin{aligned}G = \biggl( \bigl( \frac{f-gh}{p} \bigr) b \bmod g \biggr) p + g\\H = \biggl( \bigl( \frac{f-gh}{p} \bigr) a \bmod h \biggr) p + h\\B = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) b \bmod g \biggr) p + b\\A = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) a \bmod h \biggr) p + a\end{aligned}\end{align} \]

Upon return we have \(A G + B H = 1 \pmod P\) and \(f = G H \pmod P\), where \(G = g \pmod p\) etc.

We require that \(1 < p_1 \leq p\) and that the input polynomials \(f, g, h\) have degree at least \(1\) and that the input polynomials \(a\) and \(b\) are non-zero.

The output arguments \(G, H, A, B\) may only be aliased with the input arguments \(g, h, a, b\), respectively.

void fmpz_poly_hensel_lift_without_inverse(fmpz_poly_t Gout, fmpz_poly_t Hout, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

Given polynomials such that \(f = gh \pmod p\) and \(ag + bh = 1 \pmod p\), lifts only the factors \(g\) and \(h\) modulo \(P = p p_1\).

See fmpz_poly_hensel_lift().

void fmpz_poly_hensel_lift_only_inverse(fmpz_poly_t Aout, fmpz_poly_t Bout, const fmpz_poly_t G, const fmpz_poly_t H, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

Given polynomials such that \(f = gh \pmod p\) and \(ag + bh = 1 \pmod p\), lifts only the cofactors \(a\) and \(b\) modulo \(P = p p_1\).

See fmpz_poly_hensel_lift().

void fmpz_poly_hensel_lift_tree_recursive(slong *link, fmpz_poly_t *v, fmpz_poly_t *w, fmpz_poly_t f, slong j, slong inv, const fmpz_t p0, const fmpz_t p1)

Takes a current Hensel tree (link, v, w) and a pair \((j,j+1)\) of entries in the tree and lifts the tree from mod \(p_0\) to mod \(P = p_0 p_1\), where \(1 < p_1 \leq p_0\).

Set inv to \(-1\) if restarting Hensel lifting, \(0\) if stopping and \(1\) otherwise.

Here \(f = g h\) is the polynomial whose factors we are trying to lift. We will have that v[j] is the product of v[link[j]] and v[link[j] + 1] as described above.

Does support aliasing of \(f\) with one of the polynomials in the lists \(v\) and \(w\). But the polynomials in these two lists are not allowed to be aliases of each other.

void fmpz_poly_hensel_lift_tree(slong *link, fmpz_poly_t *v, fmpz_poly_t *w, fmpz_poly_t f, slong r, const fmpz_t p, slong e0, slong e1, slong inv)

Computes \(p_0 = p^{e_0}\) and \(p_1 = p^{e_1 - e_0}\) for a small prime \(p\) and \(P = p^{e_1}\).

If we aim to lift to \(p^b\) then \(f\) is the polynomial whose factors we wish to lift, made monic mod \(p^b\). As usual, (link, v, w) is an initialised tree.

This starts the recursion on lifting the product tree for lifting from \(p^{e_0}\) to \(p^{e_1}\). The value of inv corresponds to that given for the function fmpz_poly_hensel_lift_tree_recursive(). We set \(r\) to the number of local factors of \(f\).

In terms of the notation, above \(P = p^{e_1}\), \(p_0 = p^{e_0}\) and \(p_1 = p^{e_1-e_0}\).

Assumes that \(f\) is monic.

Assumes that \(1 < p_1 \leq p_0\), that is, \(0 < e_1 \leq e_0\).

slong _fmpz_poly_hensel_start_lift(fmpz_poly_factor_t lifted_fac, slong *link, fmpz_poly_t *v, fmpz_poly_t *w, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)

This function takes the local factors in local_fac and Hensel lifts them until they are known mod \(p^N\), where \(N \geq 1\).

These lifted factors will be stored (in the same ordering) in lifted_fac. It is assumed that link, v, and w are initialized arrays of fmpz_poly_t’s with at least \(2*r - 2\) entries and that \(r \geq 2\). This is done outside of this function so that you can keep them for restarting Hensel lifting later. The product of local factors must be squarefree.

The return value is an exponent which must be passed to the function _fmpz_poly_hensel_continue_lift() as prev_exp if the Hensel lifting is to be resumed.

Currently, supports the case when \(N = 1\) for convenience, although it is preferable in this case to simply iterate over the local factors and convert them to polynomials over \(\mathbf{Z}\).

slong _fmpz_poly_hensel_continue_lift(fmpz_poly_factor_t lifted_fac, slong *link, fmpz_poly_t *v, fmpz_poly_t *w, const fmpz_poly_t f, slong prev, slong curr, slong N, const fmpz_t p)

This function restarts a stopped Hensel lift.

It lifts from curr to \(N\). It also requires prev (to lift the cofactors) given as the return value of the function _fmpz_poly_hensel_start_lift() or the function _fmpz_poly_hensel_continue_lift(). The current lifted factors are supplied in lifted_fac and upon return are updated there. As usual link, v, and w describe the current Hensel tree, \(r\) is the number of local factors and \(p\) is the small prime modulo whose power we are lifting to. It is required that curr be at least \(1\) and that N > curr.

Currently, supports the case when prev and curr are equal.

void fmpz_poly_hensel_lift_once(fmpz_poly_factor_t lifted_fac, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)

This function does a Hensel lift.

It lifts local factors stored in local_fac of \(f\) to \(p^N\), where \(N \geq 2\). The lifted factors will be stored in lifted_fac. This lift cannot be restarted. This function is a convenience function intended for end users. The product of local factors must be squarefree.

Input and output

The functions in this section are not intended to be particularly fast. They are intended mainly as a debugging aid.

For the string output functions there are two variants. The first uses a simple string representation of polynomials which prints only the length of the polynomial and the integer coefficients, whilst the latter variant, appended with _pretty, uses a more traditional string representation of polynomials which prints a variable name as part of the representation.

The first string representation is given by a sequence of integers, in decimal notation, separated by white space. The first integer gives the length of the polynomial; the remaining integers are the coefficients. For example \(5x^3 - x + 1\) is represented by the string "4  1 -1 0 5", and the zero polynomial is represented by "0". The coefficients may be signed and arbitrary precision.

The string representation of the functions appended by _pretty includes only the non-zero terms of the polynomial, starting with the one of highest degree. Each term starts with a coefficient, prepended with a sign, followed by the character *, followed by a variable name, which must be passed as a string parameter to the function, followed by a caret ^ followed by a non-negative exponent.

If the sign of the leading coefficient is positive, it is omitted. Also the exponents of the degree 1 and 0 terms are omitted, as is the variable and the * character in the case of the degree 0 coefficient. If the coefficient is plus or minus one, the coefficient is omitted, except for the sign.

Some examples of the _pretty representation are:

5*x^3+7*x-4
x^2+3
-x^4+2*x-1
x+1
5

int _fmpz_poly_print(const fmpz *poly, slong len)

Prints the polynomial (poly, len) to stdout.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fmpz_poly_print(const fmpz_poly_t poly)

Prints the polynomial to stdout.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fmpz_poly_print_pretty(const fmpz *poly, slong len, const char *x)

Prints the pretty representation of (poly, len) to stdout, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fmpz_poly_print_pretty(const fmpz_poly_t poly, const char *x)

Prints the pretty representation of poly to stdout, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fmpz_poly_fprint(FILE *file, const fmpz *poly, slong len)

Prints the polynomial (poly, len) to the stream file.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fmpz_poly_fprint(FILE *file, const fmpz_poly_t poly)

Prints the polynomial to the stream file.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int _fmpz_poly_fprint_pretty(FILE *file, const fmpz *poly, slong len, const char *x)

Prints the pretty representation of (poly, len) to the stream file, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fmpz_poly_fprint_pretty(FILE *file, const fmpz_poly_t poly, const char *x)

Prints the pretty representation of poly to the stream file, using the string x to represent the indeterminate.

In case of success, returns a positive value. In case of failure, returns a non-positive value.

int fmpz_poly_read(fmpz_poly_t poly)

Reads a polynomial from stdin, storing the result in poly.

In case of success, returns a positive number. In case of failure, returns a non-positive value.

int fmpz_poly_read_pretty(fmpz_poly_t poly, char **x)

Reads a polynomial in pretty format from stdin.

For further details, see the documentation for the function fmpz_poly_fread_pretty().

int fmpz_poly_fread(FILE *file, fmpz_poly_t poly)

Reads a polynomial from the stream file, storing the result in poly.

In case of success, returns a positive number. In case of failure, returns a non-positive value.

int fmpz_poly_fread_pretty(FILE *file, fmpz_poly_t poly, char **x)

Reads a polynomial from the file file and sets poly to this polynomial. The string *x is set to the variable name that is used in the input.

Returns a positive value, equal to the number of characters read from the file, in case of success. Returns a non-positive value in case of failure, which could either be a read error or the indicator of a malformed input.

Modular reduction and reconstruction

void fmpz_poly_get_nmod_poly(nmod_poly_t Amod, const fmpz_poly_t A)

Sets the coefficients of Amod to the coefficients in A, reduced by the modulus of Amod.

void fmpz_poly_set_nmod_poly(fmpz_poly_t A, const nmod_poly_t Amod)

Sets the coefficients of A to the residues in Amod, normalised to the interval \(-m/2 \le r < m/2\) where \(m\) is the modulus.

void fmpz_poly_set_nmod_poly_unsigned(fmpz_poly_t A, const nmod_poly_t Amod)

Sets the coefficients of A to the residues in Amod, normalised to the interval \(0 \le r < m\) where \(m\) is the modulus.

void _fmpz_poly_CRT_ui_precomp(fmpz *res, const fmpz *poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, fmpz_t m1m2, ulong c, int sign)

Sets the coefficients in res to the CRT reconstruction modulo \(m_1m_2\) of the residues (poly1, len1) and (poly2, len2) which are images modulo \(m_1\) and \(m_2\) respectively. The caller must supply the precomputed product of the input moduli as \(m_1m_2\), the inverse of \(m_1\) modulo \(m_2\) as \(c\), and the precomputed inverse of \(m_2\) (in the form computed by n_preinvert_limb) as m2inv.

If sign = 0, residues \(0 \le r < m_1 m_2\) are computed, while if sign = 1, residues \(-m_1 m_2/2 \le r < m_1 m_2/2\) are computed.

Coefficients of res are written up to the maximum of len1 and len2.

void _fmpz_poly_CRT_ui(fmpz *res, const fmpz *poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, int sign)

This function is identical to _fmpz_poly_CRT_ui_precomp, apart from automatically computing \(m_1m_2\) and \(c\). It also aborts if \(c\) cannot be computed.

void fmpz_poly_CRT_ui(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_t m, const nmod_poly_t poly2, int sign)

Given poly1 with coefficients modulo m and poly2 with modulus \(n\), sets res to the CRT reconstruction modulo \(mn\) with coefficients satisfying \(-mn/2 \le c < mn/2\) (if sign = 1) or \(0 \le c < mn\) (if sign = 0).

Products

void _fmpz_poly_product_roots_fmpz_vec(fmpz *poly, const fmpz *xs, slong n)

Sets (poly, n + 1) to the monic polynomial which is the product of \((x - x_0)(x - x_1) \cdots (x - x_{n-1})\), the roots \(x_i\) being given by xs.

Aliasing of the input and output is not allowed.

void fmpz_poly_product_roots_fmpz_vec(fmpz_poly_t poly, const fmpz *xs, slong n)

Sets poly to the monic polynomial which is the product of \((x - x_0)(x - x_1) \cdots (x - x_{n-1})\), the roots \(x_i\) being given by xs.

void _fmpz_poly_product_roots_fmpq_vec(fmpz *poly, const fmpq *xs, slong n)

Sets (poly, n + 1) to the product of \((q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})\), the roots \(p_i/q_i\) being given by xs.

void fmpz_poly_product_roots_fmpq_vec(fmpz_poly_t poly, const fmpq *xs, slong n)

Sets poly to the polynomial which is the product of \((q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})\), the roots \(p_i/q_i\) being given by xs.

Roots

void _fmpz_poly_bound_roots(fmpz_t bound, const fmpz *poly, slong len)

void fmpz_poly_bound_roots(fmpz_t bound, const fmpz_poly_t poly)

Computes a nonnegative integer bound that bounds the absolute value of all complex roots of poly. Uses Fujiwara’s bound

\[2 \max \left( \left|\frac{a_{n-1}}{a_n}\right|, \left|\frac{a_{n-2}}{a_n}\right|^{\frac{1}{2}}, \dotsc, \left|\frac{a_1}{a_n}\right|^{\frac{1}{n-1}}, \left|\frac{a_0}{2a_n}\right|^{\frac{1}{n}} \right)\]

where the coefficients of the polynomial are \(a_0, \ldots, a_n\).

void _fmpz_poly_num_real_roots_sturm(slong *n_neg, slong *n_pos, const fmpz *pol, slong len)

Sets n_neg and n_pos to the number of negative and positive roots of the polynomial (pol, len) using Sturm sequence. The Sturm sequence is computed via subresultant remainders obtained by repeated call to the function _fmpz_poly_pseudo_rem_cohen.

The polynomial is assumed to be squarefree, of degree larger than 1 and with non-zero constant coefficient.

slong fmpz_poly_num_real_roots_sturm(const fmpz_poly_t pol)

Returns the number of real roots of the squarefree polynomial pol using Sturm sequence.

The polynomial is assumed to be squarefree.

slong _fmpz_poly_num_real_roots(const fmpz *pol, slong len)

Returns the number of real roots of the squarefree polynomial (pol, len).

The polynomial is assumed to be squarefree.

slong fmpz_poly_num_real_roots(const fmpz_poly_t pol)

Returns the number of real roots of the squarefree polynomial pol.

The polynomial is assumed to be squarefree.

Minimal polynomials

void _fmpz_poly_cyclotomic(fmpz *a, ulong n, nn_ptr factors, slong num_factors, ulong phi)

Sets a to the lower half of the cyclotomic polynomial \(\Phi_n(x)\), given \(n \ge 3\) which must be squarefree.

A precomputed array containing the prime factors of \(n\) must be provided, as well as the value of the Euler totient function \(\phi(n)\) as phi. If \(n\) is even, 2 must be the first factor in the list.

The degree of \(\Phi_n(x)\) is exactly \(\phi(n)\). Only the low \((\phi(n) + 1) / 2\) coefficients are written; the high coefficients can be obtained afterwards by copying the low coefficients in reverse order, since \(\Phi_n(x)\) is a palindrome for \(n \ne 1\).

We use the sparse power series algorithm described as Algorithm 4 [ArnoldMonagan2011]. The algorithm is based on the identity

\[\Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)}.\]

Treating the polynomial as a power series, the multiplications and divisions can be done very cheaply using repeated additions and subtractions. The complexity is \(O(2^k \phi(n))\) where \(k\) is the number of prime factors in \(n\).

To improve efficiency for small \(n\), we treat the fmpz coefficients as machine integers when there is no risk of overflow. The following bounds are given in Table 6 of [ArnoldMonagan2011]:

For \(n < 10163195\), the largest coefficient in any \(\Phi_n(x)\) has 27 bits, so machine arithmetic is safe on 32 bits.

For \(n < 169828113\), the largest coefficient in any \(\Phi_n(x)\) has 60 bits, so machine arithmetic is safe on 64 bits.

Further, the coefficients are always \(\pm 1\) or 0 if there are exactly two prime factors, so in this case machine arithmetic can be used as well.

Finally, we handle two special cases: if there is exactly one prime factor \(n = p\), then \(\Phi_n(x) = 1 + x + x^2 + \ldots + x^{n-1}\), and if \(n = 2m\), we use \(\Phi_n(x) = \Phi_m(-x)\) to fall back to the case when \(n\) is odd.

void fmpz_poly_cyclotomic(fmpz_poly_t poly, ulong n)

Sets poly to the \(n\)-th cyclotomic polynomial, defined as \(\Phi_n(x) = \prod_{\omega} (x-\omega)\) where \(\omega\) runs over all the \(n\)-th primitive roots of unity.

We factor \(n\) into \(n = qs\) where \(q\) is squarefree, and compute \(\Phi_q(x)\). Then \(\Phi_n(x) = \Phi_q(x^s)\).

ulong _fmpz_poly_is_cyclotomic(const fmpz *poly, slong len)

ulong fmpz_poly_is_cyclotomic(const fmpz_poly_t poly)

If poly is a cyclotomic polynomial, returns the index \(n\) of this cyclotomic polynomial. If poly is not a cyclotomic polynomial, returns 0.

void _fmpz_poly_cos_minpoly(fmpz *coeffs, ulong n)

void fmpz_poly_cos_minpoly(fmpz_poly_t poly, ulong n)

Sets poly to the minimal polynomial of \(2 \cos(2 \pi / n)\). For suitable choice of \(n\), this gives the minimal polynomial of \(2 \cos(a \pi)\) or \(2 \sin(a \pi)\) for any rational \(a\).

The cosine is multiplied by a factor two since this gives a monic polynomial with integer coefficients. One can obtain the minimal polynomial for \(\cos(2 \pi / n)\) by making the substitution \(x \to x / 2\).

For \(n > 2\), the degree of the polynomial is \(\varphi(n) / 2\). For \(n = 1, 2\), the degree is 1. For \(n = 0\), we define the output to be the constant polynomial 1.

See [WaktinsZeitlin1993].

void _fmpz_poly_swinnerton_dyer(fmpz *coeffs, ulong n)

void fmpz_poly_swinnerton_dyer(fmpz_poly_t poly, ulong n)

Sets poly to the Swinnerton-Dyer polynomial \(S_n\), defined as the integer polynomial \(S_n = \prod (x \pm \sqrt{2} \pm \sqrt{3} \pm \sqrt{5} \pm \ldots \pm \sqrt{p_n})\) where \(p_n\) denotes the \(n\)-th prime number and all combinations of signs are taken. This polynomial has degree \(2^n\) and is irreducible over the integers (it is the minimal polynomial of \(\sqrt{2} + \ldots + \sqrt{p_n}\)).

Orthogonal polynomials

void _fmpz_poly_chebyshev_t(fmpz *coeffs, ulong n)

void fmpz_poly_chebyshev_t(fmpz_poly_t poly, ulong n)

Sets poly to the Chebyshev polynomial of the first kind \(T_n(x)\), defined by \(T_n(x) = \cos(n \cos^{-1}(x))\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence.

void _fmpz_poly_chebyshev_u(fmpz *coeffs, ulong n)

void fmpz_poly_chebyshev_u(fmpz_poly_t poly, ulong n)

Sets poly to the Chebyshev polynomial of the first kind \(U_n(x)\), defined by \((n+1) U_n(x) = T'_{n+1}(x)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence.

void _fmpz_poly_legendre_pt(fmpz *coeffs, ulong n)

Sets coeffs to the coefficient array of the shifted Legendre polynomial \(\tilde{P_n}(x)\), defined by \(\tilde{P_n}(x) = P_n(2x-1)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence. The length of the array will be n+1. See fmpq_poly for the Legendre polynomials.

void fmpz_poly_legendre_pt(fmpz_poly_t poly, ulong n)

Sets poly to the shifted Legendre polynomial \(\tilde{P_n}(x)\), defined by \(\tilde{P_n}(x) = P_n(2x-1)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence. See fmpq_poly for the Legendre polynomials.

void _fmpz_poly_hermite_h(fmpz *coeffs, ulong n)

Sets coeffs to the coefficient array of the Hermite polynomial \(H_n(x)\), defined by \(H'_n(x) = 2nH_{n-1}(x)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence. The length of the array will be n+1.

void fmpz_poly_hermite_h(fmpz_poly_t poly, ulong n)

Sets poly to the Hermite polynomial \(H_n(x)\), defined by \(H'_n(x) = 2nH_{n-1}(x)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence.

void _fmpz_poly_hermite_he(fmpz *coeffs, ulong n)

Sets coeffs to the coefficient array of the Hermite polynomial \(He_n(x)\), defined by \(He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence. The length of the array will be n+1.

void fmpz_poly_hermite_he(fmpz_poly_t poly, ulong n)

Sets poly to the Hermite polynomial \(He_n(x)\), defined by \(He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)\), for \(n\ge0\). The coefficients are calculated using a hypergeometric recurrence.

Fibonacci polynomials

void _fmpz_poly_fibonacci(fmpz *coeffs, ulong n)

Sets coeffs to the coefficient array of the \(n\)-th Fibonacci polynomial. The coefficients are calculated using a hypergeometric recurrence.

void fmpz_poly_fibonacci(fmpz_poly_t poly, ulong n)

Sets poly to the \(n\)-th Fibonacci polynomial. The coefficients are calculated using a hypergeometric recurrence.

Eulerian numbers and polynomials

Eulerian numbers are the coefficients to the Eulerian polynomials

\[A_n(x) = \sum_{m = 0}^{n} A(n, m) x^m,\]

where the Eulerian polynomials are defined by the exponential generating function

\[\frac{x - 1}{x - e^{(x - 1) t}} = \sum_{n = 0}^{\infty} A_n(x) \frac{t^n}{n!}.\]

The Eulerian numbers can be expressed explicitly via the formula

\[A(n, m) = \sum_{k = 0}^{m + 1} (-1)^k \binom{n + 1}{k} (m + 1 - k)^n.\]

Note: Not to be confused with Euler numbers and polynomials.

void fmpz_eulerian_polynomial(fmpz_poly_t res, ulong n)

Sets res to the Eulerian polynomial \(A_n(x)\), where we define \(A_0(x) = 1\). The polynomial is calculated via a recursive relation.

Modular forms and q-series

void _fmpz_poly_eta_qexp(fmpz *f, slong r, slong len)

void fmpz_poly_eta_qexp(fmpz_poly_t f, slong r, slong n)

Sets \(f\) to the \(q\)-expansion to length \(n\) of the Dedekind eta function (without the leading factor \(q^{1/24}\)) raised to the power \(r\), i.e. \((q^{-1/24} \eta(q))^r = \prod_{k=1}^{\infty} (1 - q^k)^r\).

In particular, \(r = -1\) gives the generating function of the partition function \(p(k)\), and \(r = 24\) gives, after multiplication by \(q\), the modular discriminant \(\Delta(q)\) which generates the Ramanujan tau function \(\tau(k)\).

This function uses sparse formulas for \(r = 1, 2, 3, 4, 6\) and otherwise reduces to one of those cases using power series arithmetic.

void _fmpz_poly_theta_qexp(fmpz *f, slong r, slong len)

void fmpz_poly_theta_qexp(fmpz_poly_t f, slong r, slong n)

Sets \(f\) to the \(q\)-expansion to length \(n\) of the Jacobi theta function raised to the power \(r\), i.e. \(\vartheta(q)^r\) where \(\vartheta(q) = 1 + 2 \sum_{k=1}^{\infty} q^{k^2}\).

This function uses sparse formulas for \(r = 1, 2\) and otherwise reduces to those cases using power series arithmetic.

CLD bounds

void fmpz_poly_CLD_bound(fmpz_t res, const fmpz_poly_t f, slong n)

Compute a bound on the \(n\) coefficient of \(fg'/g\) where \(g\) is any factor of \(f\).