qfb.h – binary quadratic forms¶
Authors:
William Hart
Memory management¶
-
void qfb_init(qfb_t q)¶
Initialise a code{qfb_t} \(q\) for use.
-
void qfb_clear(qfb_t q)¶
Clear a code{qfb_t} after use. This releases any memory allocated for \(q\) back to flint.
Hash table¶
-
void qfb_hash_clear(qfb_hash_t *qhash, slong depth)¶
Frees all memory used by a hash table of size \(2^{depth}\).
-
void qfb_hash_insert(qfb_hash_t *qhash, qfb_t q, qfb_t q2, slong iter, slong depth)¶
Insert the binary quadratic form code{q} into the given hash table of size \(2^{depth}\) in the field code{q} of the hash structure. Also store the second binary quadratic form code{q2} (if not code{NULL}) in the similarly named field and code{iter} in the similarly named field of the hash structure.
Basic manipulation¶
-
void qfb_set(qfb_t f, qfb_t g)¶
Set the binary quadratic form \(f\) to be equal to \(g\).
Comparison¶
-
int qfb_equal(qfb_t f, qfb_t g)¶
Returns \(1\) if \(f\) and \(g\) are identical binary quadratic forms, otherwise returns \(0\).
Input/output¶
-
void qfb_print(qfb_t q)¶
Print a binary quadratic form \(q\) in the format \((a, b, c)\) where \(a\), \(b\), \(c\) are the entries of \(q\).
Computing with forms¶
-
void qfb_discriminant(fmpz_t D, qfb_t f)¶
Set \(D\) to the discriminant of the binary quadratic form \(f\), i.e. to \(b^2 - 4ac\), where \(f = (a, b, c)\).
-
void qfb_reduce(qfb_t r, qfb_t f, fmpz_t D)¶
Set \(r\) to the reduced form equivalent to the binary quadratic form \(f\) of discriminant \(D\).
-
int qfb_is_reduced(qfb_t r)¶
Returns \(1\) if \(q\) is a reduced binary quadratic form. Otherwise returns \(1\).
-
slong qfb_reduced_forms(qfb **forms, slong d)¶
Given a discriminant \(d\) (negative for negative definite forms), compute all the reduced binary quadratic forms of that discriminant. The function allocates space for these and returns it in the variable code{forms} (the user is responsible for cleaning this up by a single call to code{qfb_array_clear} on code{forms}, after use. The function returns the number of forms generated (the form class number). The forms are stored in an array of code{qfb} structs, which contain fields code{a, b, c} corresponding to forms \((a, b, c)\).
-
slong qfb_reduced_forms_large(qfb **forms, slong d)¶
As for
qfb_reduced_forms. However, for small \(|d|\) it requires fewer primes to be computed at a small cost in speed. It is called automatically by code{qfb_reduced_forms} for large \(|d|\) so thatflint_primesis not exhausted.
-
void qfb_nucomp(qfb_t r, qfb_t f, qfb_t g, fmpz_t D, fmpz_t L)¶
Shanks’ NUCOMP as described in~citep{JacvdP}
% Computational aspects of NUCOMP”, Michael J. Jacobson Jr., % Alfred J. van der Poorten, ANTS 2002, LNCS 2369, pp. 120–133.
Computes the near reduced composition of forms \(f\) and \(g\) given \(L = \lfloor |D|^{1/4} \rfloor\) where \(D\) is the common discriminant of \(f\) and \(g\). The result is returned in \(r\).
We require that that \(f\) is a primitive form.
-
void qfb_nudupl(qfb_t r, qfb_t f, fmpz_t D, fmpz_t L)¶
As for code{nucomp} except that the form \(f\) is composed with itself. We require that that \(f\) is a primitive form.
-
void qfb_pow_ui(qfb_t r, qfb_t f, fmpz_t D, ulong exp)¶
Compute the near reduced form \(r\) which is the result of composing the principal form (identity) with \(f\) code{exp} times.
We require \(D\) to be set to the discriminant of \(f\) and that \(f\) is a primitive form.
-
void qfb_inverse(qfb_t r, qfb_t f)¶
Set \(r\) to the inverse of the binary quadratic form \(f\).
-
int qfb_is_principal_form(qfb_t f, fmpz_t D)¶
Return \(1\) if \(f\) is the reduced principal form of discriminant \(D\), i.e. the identity in the form class group.
-
void qfb_principal_form(qfb_t f, fmpz_t D)¶
Set \(f\) to the principal form of discriminant \(D\), i.e. the identity in the form class group.
-
int qfb_is_primitive(qfb_t f)¶
Return \(1\) if \(f\) is primitive, i.e. the greatest common divisor of its three coefficients is \(1\). Otherwise the function returns \(0\).
-
void qfb_prime_form(qfb_t r, fmpz_t D, fmpz_t p)¶
Sets \(r\) to the unique prime \((p, b, c)\) of discriminant \(D\), i.e. with \(0 < b \leq p\). We require that \(p\) is a prime.
-
int qfb_exponent_element(fmpz_t exponent, qfb_t f, fmpz_t n, ulong B1, ulong B2_sqrt)¶
Find the exponent of the element \(f\) in the form class group of forms of discriminant \(n\), doing a stage \(1\) with primes up to at least code{B1} and a stage \(2\) for a single large prime up to at least the square of code{B2}. If the function fails to find the exponent it returns \(0\), otherwise the function returns \(1\) and code{exponent} is set to the exponent of \(f\), i.e. the minimum power of \(f\) which gives the identity.
It is assumed that the form \(f\) is reduced. We require that code{iters} is a power of \(2\) and that code{iters}` >= 1024`.
The function performs a stage \(2\) which stores up to \(4\times\) code{iters} binary quadratic forms, and \(12\times\) code{iters} additional limbs of data in a hash table, where code{iters} is the square root of code{B2}.
-
int qfb_exponent(fmpz_t exponent, fmpz_t n, ulong B1, ulong B2_sqrt, slong c)¶
Compute the exponent of the class group of discriminant \(n\), doing a stage \(1\) with primes up to at least code{B1} and a stage \(2\) for a single large prime up to at least the square of code{B2_sqrt}, and with probability at least \(1 - 2^{-c}\). If the prime limits are exhausted without finding the exponent, the function returns \(0\), otherwise it returns \(1\) and code{exponent} is set to the computed exponent, i.e. the minimum power which every element of the class group has to be raised to give the identity.
The function performs a stage \(2\) which stores up to \(4\times\) code{iters} binary quadratic forms, and \(12\times\) code{iters} additional limbs of data in a hash table, where code{iters} is the square root of code{B2}.
We use algorithm 8.1 of~citep{SuthThesis}
% “Order Computations in Generic Groups”, Andrew Sutherland, % MIT Thesis 2007. % http://groups.csail.mit.edu/cis/theses/sutherland-phd.pdf
-
int qfb_exponent_grh(fmpz_t exponent, fmpz_t n, ulong iters, ulong B1, ulong B2_sqrt)¶
As per code{qfb_exponent} except that the bound code{c} is automatically generated such that the exponent it guaranteed to be correct, if found, assuming the GRH, namely that the class group is generated by primes less than \(6\log^2(|n|)\) as per~citep{BuchDull1992}
% “Distributed Class Group Computation”, Johannes Buchmann, Stephan % D"{u}llman, Informatik 1 (1992), pp. 69–79.