mpoly.h – support functions for multivariate polynomials¶
An array of type
ulong *
orfmpz **
is used to communicate exponent vectors. These exponent vectors must have length equal to the number of variables in the polynomial ring. The element of this exponent vector at index \(0\) corresponds to the most significant variable in the monomial ordering. For example, if the polynomial is \(7\cdot x^2\cdot y+8\cdot y\cdot z+9\) and the variables are ordered so that \(x>y>z\), the degree function will return \(\{2,1,1\}\). Similarly, the exponent vector of the \(0\)index term of this polynomial is \(\{2,1,0\}\), while the \(2\)index term has exponent vector \(\{0,0,0\}\) and coefficient \(9\).
Orderings¶

type ordering_t¶
Represents one of the following supported term orderings:

ORD_LEX¶
The lexicographic ordering.

ORD_DEGLEX¶
The degree lexicographic ordering.

ORD_DEGREVLEX¶
The degree reverse lexicographic ordering.

ORD_LEX¶

type mpoly_ctx_struct¶

type mpoly_ctx_t¶
An mpoly_ctx_struct is a structure holding information about the number of variables and the term ordering of a multivariate polynomial.

void mpoly_ctx_init(mpoly_ctx_t ctx, slong nvars, const ordering_t ord)¶
Initialize a context for specified number of variables and ordering.

void mpoly_ctx_clear(mpoly_ctx_t mctx)¶
Clean up any space used by a context object.

ordering_t mpoly_ordering_randtest(flint_rand_t state)¶
Return a random term ordering.

void mpoly_ctx_init_rand(mpoly_ctx_t mctx, flint_rand_t state, slong max_nvars)¶
Initialize a context with a random choice for the ordering.

int mpoly_ordering_isdeg(const mpoly_ctx_t ctx)¶
Return 1 if the ordering of the given context is a degree ordering (deglex or degrevlex).

int mpoly_ordering_isrev(const mpoly_ctx_t cth)¶
Return 1 if the ordering of the given context is a reverse ordering (currently only degrevlex).

void mpoly_ordering_print(ordering_t ord)¶
Print a string (either “lex”, “deglex” or “degrevlex”) to standard output, corresponding to the given ordering.
Monomial arithmetic¶
These functions in this section are only provided as inline functions as they are somewhat trivial. This is in order to minimize the FLINT binary.

void mpoly_monomial_add(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N)¶
Set
(exp_ptr, N)
to the sum of the monomials(exp2, N)
and(exp3, N)
, assumingbits <= FLINT_BITS
.

void mpoly_monomial_add_mp(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N)¶
Set
(exp_ptr, N)
to the sum of the monomials(exp2, N)
and(exp3, N)
.

void mpoly_monomial_sub(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N)¶
Set
(exp_ptr, N)
to the difference of the monomials(exp2, N)
and(exp3, N)
, assumingbits <= FLINT_BITS

void mpoly_monomial_sub_mp(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N)¶
Set
(exp_ptr, N)
to the difference of the monomials(exp2, N)
and(exp3, N)
.

int mpoly_monomial_overflows(ulong *exp2, slong N, ulong mask)¶
Return true if any of the fields of the given monomial
(exp2, N)
has overflowed (or is negative). Themask
is a word with the high bit of each field set to 1. In other words, the function returns 1 if any word ofexp2
has any of the nonzero bits inmask
set. Assumes thatbits <= FLINT_BITS
.

int mpoly_monomial_overflows_mp(ulong *exp_ptr, slong N, flint_bitcnt_t bits)¶
Return true if any of the fields of the given monomial
(exp_ptr, N)
has overflowed. Assumes thatbits >= FLINT_BITS
.

void mpoly_monomial_set(ulong *exp2, const ulong *exp3, slong N)¶
Set the monomial
(exp2, N)
to(exp3, N)
.
Monomial comparison¶
These functions in this section are only provided as inline functions as they are somewhat trivial. This is in order to minimize the FLINT binary.

int mpoly_monomial_equal(const ulong *exp2, const ulong *exp3, slong N)¶
Return 1 if the monomials
(exp2, N)
and(exp3, N)
are equal.

void mpoly_get_cmpmask(ulong *cmpmask, slong N, ulong bits, const mpoly_ctx_t mctx)¶
Get the mask
(cmpmask, N)
for comparisons.bits
should be set to the number of bits in the exponents to be compared. Any function that compares monomials should use this comparison mask.

int mpoly_monomial_lt(const ulong *exp2, const ulong *exp3, slong N, const ulong *cmpmask)¶
Return 1 if
(exp2, N)
is less than(exp3, N)
.
Monomial divisibility¶
These functions in this section are only provided as inline functions as they are somewhat trivial. This is in order to minimize the FLINT binary.

int mpoly_monomial_divides(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N, ulong mask)¶
Return 1 if the monomial
(exp3, N)
divides(exp2, N)
. If so set(exp_ptr, N)
to the quotient monomial. Themask
is a word with the high bit of each bit field set to 1. Assumes thatbits <= FLINT_BITS
.

int mpoly_monomial_divides_mp(ulong *exp_ptr, const ulong *exp2, const ulong *exp3, slong N, flint_bitcnt_t bits)¶
Return 1 if the monomial
(exp3, N)
divides(exp2, N)
. If so set(exp_ptr, N)
to the quotient monomial. Assumes thatbits >= FLINT_BITS
.

int mpoly_monomial_divides1(ulong *exp_ptr, const ulong exp2, const ulong exp3, ulong mask)¶
As per
mpoly_monomial_divides
withN = 1
.

int mpoly_monomial_divides_tight(slong e1, slong e2, slong *prods, slong num)¶
Return 1 if the monomial
e2
divides the monomiale1
, where the monomials are stored using factorial representation. The array(prods, num)
should consist of \(1\), \(b_1, b_1\times b_2, \ldots\), where the \(b_i\) are the bases of the factorial number representation.
Basic manipulation¶

flint_bitcnt_t mpoly_exp_bits_required_ui(const ulong *user_exp, const mpoly_ctx_t mctx)¶
Returns the number of bits required to store
user_exp
in packed format. The returned number of bits includes space for a zeroed signed bit.

flint_bitcnt_t mpoly_exp_bits_required_ffmpz(const fmpz *user_exp, const mpoly_ctx_t mctx)¶
Returns the number of bits required to store
user_exp
in packed format. The returned number of bits includes space for a zeroed signed bit.

flint_bitcnt_t mpoly_exp_bits_required_pfmpz(fmpz *const *user_exp, const mpoly_ctx_t mctx)¶
Returns the number of bits required to store
user_exp
in packed format. The returned number of bits includes space for a zeroed signed bit.

void mpoly_max_fields_ui_sp(ulong *max_fields, const ulong *poly_exps, slong len, ulong bits, const mpoly_ctx_t mctx)¶
Compute the fieldwise maximum of packed exponents from
poly_exps
of lengthlen
and unpack the result intomax_fields
. The maximums are assumed to fit a ulong.

void mpoly_max_fields_fmpz(fmpz *max_fields, const ulong *poly_exps, slong len, ulong bits, const mpoly_ctx_t mctx)¶
Compute the fieldwise maximum of packed exponents from
poly_exps
of lengthlen
and unpack the result intomax_fields
.

void mpoly_max_degrees_tight(slong *max_exp, ulong *exps, slong len, slong *prods, slong num)¶
Return an array of
num
integers corresponding to the maximum degrees of the exponents in the array of exponent vectors(exps, len)
, assuming that the exponent are packed in a factorial representation. The array(prods, num)
should consist of \(1\), \(b_1\), \(b_1\times b_2, \ldots\), where the \(b_i\) are the bases of the factorial number representation. The results are stored in the arraymax_exp
, with the entry corresponding to the most significant base of the factorial representation first in the array.

int mpoly_monomial_exists(slong *index, const ulong *poly_exps, const ulong *exp, slong len, slong N, const ulong *cmpmask)¶
Returns true if the given exponent vector
exp
exists in the array of exponent vectors(poly_exps, len)
, otherwise, returns false. If the exponent vector is found, its index into the array of exponent vectors is returned. Otherwise,index
is set to the index where this exponent could be inserted to preserve the ordering. The index can be in the range[0, len]
.

void mpoly_search_monomials(slong **e_ind, ulong *e, slong *e_score, slong *t1, slong *t2, slong *t3, slong lower, slong upper, const ulong *a, slong a_len, const ulong *b, slong b_len, slong N, const ulong *cmpmask)¶
Given packed exponent vectors
a
andb
, compute a packed exponente
such that the number of monomials in the cross producta
Xb
that are less than or equal toe
is betweenlower
andupper
. This number is stored ine_store
. If no such monomial exists, one is chosen so that the number of monomials is as close as possible. This function assumes that1
is the smallest monomial and needs three arrayst1
,t2
, andt3
of the size asa
for workspace. The parametere_ind
is set to one oft1
,t2
, andt3
and gives the locations of the monomials ina
Xb
.
Setting and getting monomials¶

int mpoly_term_exp_fits_ui(ulong *exps, ulong bits, slong n, const mpoly_ctx_t mctx)¶
Return whether every entry of the exponent vector of index \(n\) in
exps
fits into aulong
.

int mpoly_term_exp_fits_si(ulong *exps, ulong bits, slong n, const mpoly_ctx_t mctx)¶
Return whether every entry of the exponent vector of index \(n\) in
exps
fits into aslong
.

void mpoly_get_monomial_ui(ulong *exps, const ulong *poly_exps, ulong bits, const mpoly_ctx_t mctx)¶
Convert the packed exponent
poly_exps
of bit countbits
to a monomial from the user’s perspective. The exponents are assumed to fit a ulong.

void mpoly_get_monomial_ffmpz(fmpz *exps, const ulong *poly_exps, flint_bitcnt_t bits, const mpoly_ctx_t mctx)¶
Convert the packed exponent
poly_exps
of bit countbits
to a monomial from the user’s perspective.

void mpoly_get_monomial_pfmpz(fmpz **exps, const ulong *poly_exps, flint_bitcnt_t bits, const mpoly_ctx_t mctx)¶
Convert the packed exponent
poly_exps
of bit countbits
to a monomial from the user’s perspective.

void mpoly_set_monomial_ui(ulong *exp1, const ulong *exp2, ulong bits, const mpoly_ctx_t mctx)¶
Convert the user monomial
exp2
to packed format usingbits
.

void mpoly_set_monomial_ffmpz(ulong *exp1, const fmpz *exp2, flint_bitcnt_t bits, const mpoly_ctx_t mctx)¶
Convert the user monomial
exp2
to packed format usingbits
.

void mpoly_set_monomial_pfmpz(ulong *exp1, fmpz *const *exp2, flint_bitcnt_t bits, const mpoly_ctx_t mctx)¶
Convert the user monomial
exp2
to packed format usingbits
.
Packing and unpacking monomials¶

void mpoly_pack_vec_ui(ulong *exp1, const ulong *exp2, ulong bits, slong nfields, slong len)¶
Packs a vector
exp2
into {exp1} using a bit count ofbits
. No checking is done to ensure that the vector actually fits intobits
bits. The number of fields in each vector isnfields
and the total number of vectors to unpack islen
.

void mpoly_pack_vec_fmpz(ulong *exp1, const fmpz *exp2, flint_bitcnt_t bits, slong nfields, slong len)¶
Packs a vector
exp2
into {exp1} using a bit count ofbits
. No checking is done to ensure that the vector actually fits intobits
bits. The number of fields in each vector isnfields
and the total number of vectors to unpack islen
.

void mpoly_unpack_vec_ui(ulong *exp1, const ulong *exp2, ulong bits, slong nfields, slong len)¶
Unpacks vector
exp2
of bit countbits
intoexp1
. The number of fields in each vector isnfields
and the total number of vectors to unpack islen
.

void mpoly_unpack_vec_fmpz(fmpz *exp1, const ulong *exp2, flint_bitcnt_t bits, slong nfields, slong len)¶
Unpacks vector
exp2
of bit countbits
intoexp1
. The number of fields in each vector isnfields
and the total number of vectors to unpack islen
.

int mpoly_repack_monomials(ulong *exps1, ulong bits1, const ulong *exps2, ulong bits2, slong len, const mpoly_ctx_t mctx)¶
Convert an array of length
len
of exponentsexps2
packed using bitsbits2
into an arrayexps1
using bitsbits1
. No checking is done to ensure that the result fits into bitsbits1
.

void mpoly_pack_monomials_tight(ulong *exp1, const ulong *exp2, slong len, const slong *mults, slong num, slong bits)¶
Given an array of possibly packed exponent vectors
exp2
of lengthlen
, where each field of each exponent vector is packed into the given number of bits, return the corresponding array of monomial vectors packed using a factorial numbering scheme. The “bases” for the factorial numbering scheme are given as an array of integersmults
, the first entry of which corresponds to the field of least significance in each input exponent vector. Obviously the maximum exponent to be packed must be less than the corresponding base inmults
.The number of multipliers is given by
num
. The code only considers least significantnum
fields of each exponent vectors and ignores the rest. The number of ignored fields should be passed inextras
.

void mpoly_unpack_monomials_tight(ulong *e1, ulong *e2, slong len, slong *mults, slong num, slong bits)¶
Given an array of exponent vectors
e2
of lengthlen
packed using a factorial numbering scheme, unpack the monomials into an arraye1
of exponent vectors in standard packed format, where each field has the given number of bits. The “bases” for the factorial numbering scheme are given as an array of integersmults
, the first entry of which corresponds to the field of least significance in each exponent vector.The number of multipliers is given by
num
. The code only considers least significantnum
fields of each exponent vectors and ignores the rest. The number of ignored fields should be passed inextras
.
Chunking¶

void mpoly_main_variable_terms1(slong *i1, slong *n1, const ulong *exp1, slong l1, slong len1, slong k, slong num, slong bits)¶
Given an array of exponent vectors
(exp1, len1)
, each exponent vector taking one word of space, with each exponent being packed into the given number of bits, computel1
starting offsetsi1
and lengthsn1
(which may be zero) to break the exponents into chunks. Each chunk consists of exponents have the same degree in the main variable. The index of the main variable is given by \(k\). The variables are indexed from the variable of least significance, starting from \(0\). The valuel1
should be the degree in the main variable, plus one.
Chained heap functions¶

int _mpoly_heap_insert(mpoly_heap_s *heap, ulong *exp, void *x, slong *next_loc, slong *heap_len, slong N, const ulong *cmpmask)¶
Given a heap, insert a new node \(x\) corresponding to the given exponent into the heap. Heap elements are ordered by the exponent
(exp, N)
, with the largest element at the head of the heap. A pointer to the current heap length must be passed in viaheap_len
. This will be updated by the function. Note that the index 0 position in the heap is not used, so the length is always one greater than the number of elements.

void _mpoly_heap_insert1(mpoly_heap1_s *heap, ulong exp, void *x, slong *next_loc, slong *heap_len, ulong maskhi)¶
As per
_mpoly_heap_insert
except thatN = 1
, andmaskhi = cmpmask[0]
.

void *_mpoly_heap_pop(mpoly_heap_s *heap, slong *heap_len, slong N, const ulong *cmpmask)¶
Pop the head of the heap. It is cast to a
void *
. A pointer to the current heap length must be passed in viaheap_len
. This will be updated by the function. Note that the index 0 position in the heap is not used, so the length is always one greater than the number of elements. Themaskhi
andmasklo
values are zero except for degrevlex ordering, where they are as per the monomial comparison operations above.