# HG changeset patch # User Gustavo Rama # Date 1316721423 10800 # Node ID 7cec1a24a55e8ddbed3b1fca0df43d4c9b2fcf4e # Parent b29133ea7a08ebd71615cabd96a41f8397551970 [mq]: ternary-fixes-1 diff -r b29133ea7a08 -r 7cec1a24a55e sage/quadratic_forms/random_quadraticform.py --- a/sage/quadratic_forms/random_quadraticform.py Thu Jun 03 23:22:13 2010 -0300 +++ b/sage/quadratic_forms/random_quadraticform.py Thu Sep 22 16:57:03 2011 -0300 @@ -121,7 +121,39 @@ ## Return the quadratic form return Q -def random_ternaryqf(rand_arg_list): +def random_ternaryqf(rand_arg_list = []): + """ + Create a random ternary quadratic form. + + The last (and optional) argument ``rand_arg_list`` is a list of at most 3 + elements which is passed (as at most 3 separate variables) into the method + ``R.random_element()``. + + INPUT: + + - ``rand_arg_list`` -- a list of at most 3 arguments which can be taken by + ``R.random_element()``. + + OUTPUT: + + A ternary quadratic form. + + EXAMPLES:: + + sage: random_ternaryqf() ##RANDOM + Ternary quadratic form with integer coefficients: + [1 1 4] + [-1 1 -1] + sage: random_ternaryqf([-1, 2]) ##RANDOM + Ternary quadratic form with integer coefficients: + [1 0 1] + [-1 -1 -1] + sage: random_ternaryqf([-10, 10, "uniform"]) ##RANDOM + Ternary quadratic form with integer coefficients: + [7 -8 2] + [0 3 -6] + """ + R = ZZ n = 6 @@ -140,6 +172,26 @@ def random_ternaryqf_with_conditions(condition_list=[], rand_arg_list=[]): + """ + Create a random ternary quadratic form satisfying a list of boolean + (i.e. True/False) conditions. + + The conditions `c` appearing in the list must be boolean functions which + can be called either as ``Q.c()`` or ``c(Q)``, where ``Q`` is the random + ternary quadratic form. + + The last (and optional) argument ``rand_arg_list`` is a list of at most 3 + elements which is passed (as at most 3 separate variables) into the method + ``R.random_element()``. + + EXAMPLES:: + + sage: Q = random_ternaryqf_with_conditions([TernaryQF.is_positive_definite], [-5, 5]) + sage: Q ## RANDOM + Ternary quadratic form with integer coefficients: + [3 4 2] + [2 -2 -1] + """ Q = random_ternaryqf(rand_arg_list) Done_Flag = True diff -r b29133ea7a08 -r 7cec1a24a55e sage/quadratic_forms/ternary.pyx --- a/sage/quadratic_forms/ternary.pyx Thu Jun 03 23:22:13 2010 -0300 +++ b/sage/quadratic_forms/ternary.pyx Thu Sep 22 16:57:03 2011 -0300 @@ -7,21 +7,36 @@ from sage.misc.prandom import randint from sage.rings.arith import xgcd,gcd from sage.functions.other import ceil, floor -from math import sqrt from __builtin__ import max def red_mfact(a,b): + """ + Auxiliar function for reduction that finds the reduction factor of a, b integers. + + INPUT: + - a, b integers + + OUTPUT: + Integer + + EXAMPLES:: + + sage: from sage.quadratic_forms.ternary import red_mfact + sage: red_mfact(0, 3) + 0 + sage: red_mfact(-5, 100) + 9 + + """ if a: - return((-b+abs(a))//(2*a)) + return (-b + abs(a))//(2*a) else: return 0 def _reduced_ternary_form_eisenstein_with_matrix(a1, a2, a3, a23, a13, a12): - - """ Find the coefficients of the equivalent unique reduced ternary form according to the conditions of Dickson's "Studies in the Theory of Numbers", pp164-171, and the tranformation matrix. @@ -223,7 +238,6 @@ return((a1,a2,a3,a23,a13,a12),M) def _reduced_ternary_form_eisenstein_without_matrix(a1, a2, a3, a23, a13, a12): - """ Find the coefficients of the equivalent unique reduced ternary form according to the conditions of Dickson's "Studies in the Theory of Numbers", pp164-171. @@ -393,7 +407,6 @@ def primitivize(long long v0, long long v1, long long v2, p): - """ Given a 3-tuple v not singular mod p, it returns a primitive 3-tuple version of v mod p. @@ -416,7 +429,6 @@ return 1, 0, 0 def evaluate(a, b, c, r, s, t, v): - """ Function to evaluate the ternary quadratic form (a, b, c, r, s, t) in a 3-tuple v. @@ -435,7 +447,6 @@ return a*v[0]**2+b*v[1]**2+c*v[2]**2+r*v[2]*v[1]+s*v[2]*v[0]+t*v[1]*v[0] def _find_zeros_mod_p_2(a, b, c, r, s, t): - """ Function to find the zeros mod 2 of a ternary quadratic form. @@ -471,7 +482,6 @@ return zeros def pseudorandom_primitive_zero_mod_p(a, b, c, r, s, t, p): - """ Find a zero of the form (a, b, 1) of the ternary quadratic form given by the coefficients (a, b, c, r, s, t) mod p, where p is a odd prime that doesn't divides the discriminant. @@ -507,7 +517,6 @@ return z%p, (r1*z+r2)%p, 1 def _find_zeros_mod_p_odd(long long a, long long b, long long c, long long r, long long s, long long t, long long p, v): - """ Find the zeros mod p, where p is an odd prime, of a ternary quadratic form given by it's coefficients and a given zero of the form v. The prime p doesn't divides the discriminant of the form. @@ -578,7 +587,6 @@ def _find_zeros_mod_p(a, b, c, r, s, t, p): - """ Finds the zeros mod p of the ternary quadratic form given by the coefficients (a, b, c, r, s, t), where p is a prime that doesn't divides the discriminant of the form. @@ -610,7 +618,6 @@ def _find_all_ternary_qf_by_level_disc(long long N, long long d): - """ Find the coefficients of all the reduced ternary quadratic forms given it's discriminant d and level N. If N|4d and d|N^2, then it may be some forms with that discriminant and level. @@ -680,7 +687,7 @@ while a<=a_max: [g,u,v]=xgcd(4*a,m) - g1=ZZ(sqrt(ZZ(g).squarefree_part()*g)) + g1=(ZZ(g).squarefree_part()*g).sqrtrem()[0] t=0 while t<=a: @@ -756,7 +763,6 @@ def _find_a_ternary_qf_by_level_disc(long long N, long long d): - """ Find the coefficients of a reduced ternary quadratic form given it's discriminant d and level N. If N|4d and d|N^2, then it may be a form with that discriminant and level. @@ -822,7 +828,7 @@ while a<=a_max: [g,u,v]=xgcd(4*a,m) - g1=ZZ(sqrt(ZZ(g).squarefree_part()*g)) + g1=(ZZ(g).squarefree_part()*g).sqrtrem()[0] t=0 while t<=a: @@ -897,7 +903,6 @@ def extend(v): - """ Return the coefficients of a matrix M such that M has determinant gcd(v) and the first column is v. @@ -933,7 +938,6 @@ def _find_p_neighbor_from_vec(a, b, c, r, s, t, p, v, mat = False): - """ Finds the coefficients of the reduced equivalent of the p-neighbor of the ternary quadratic given by Q = (a, b, c, r, s, t) form associated @@ -1042,7 +1046,6 @@ def _basic_lemma_vec(a, b, c, r, s, t, n): - """ Find a vector v such that the ternary quadratic form given by (a, b, c, r, s, t) evaluated at v is coprime with n a prime or 1. @@ -1079,7 +1082,6 @@ raise ValueError, "not primitive form" def _basic_lemma(a, b, c, r, s, t, n): - """ Finds a number represented by the ternary quadratic form given by the coefficients (a, b, c, r, s, t) and coprime to the prime n. diff -r b29133ea7a08 -r 7cec1a24a55e sage/quadratic_forms/ternary_qf.py --- a/sage/quadratic_forms/ternary_qf.py Thu Jun 03 23:22:13 2010 -0300 +++ b/sage/quadratic_forms/ternary_qf.py Thu Sep 22 16:57:03 2011 -0300 @@ -1,6 +1,12 @@ """ Ternary Quadratic Form with integer coefficients. +AUTHOR: + + - Gustavo Rama + +Based in code of Gonzalo Tornaria + The form `a*x^2 + b*y^2 + c*z^2 + r*yz + s*xz + t*xy` is stored as a tuple (a, b, c, r, s, t) of integers. """ @@ -20,20 +26,17 @@ from sage.modules.free_module_element import vector from sage.rings.ring import is_Ring from sage.rings.rational_field import QQ -#from sage.quadratic_forms.ternary import _hecke_matrix class TernaryQF(SageObject): - """ The ``TernaryQF`` class represents a quadratic form in 3 variables with coefficients in Z. INPUT: - - `v` -- a list or tuple of 6 entries: [a,b,c,r,s,t] OUTPUT: - the ternary quadratic form a*x^2 + b*y^2 + c*z^2 + r*y*z + s*x*z + t*x*y. + - the ternary quadratic form a*x^2 + b*y^2 + c*z^2 + r*y*z + s*x*z + t*x*y. EXAMPLES:: @@ -47,6 +50,7 @@ Ternary quadratic form with integer coefficients: [1 187 9] [-85 8 -31] + sage: TestSuite(TernaryQF).run() """ @@ -55,7 +59,6 @@ __slots__ = ['_a', '_b', '_c', '_r', '_s', '_t'] def __init__(self,v): - """ Creates the ternary quadratic form `a*x^2 + b*y^2 + c*z^2 + r*y*z + s*x*z + t*x*y.` from the tuple v=[a,b,c,r,s,t] over `\ZZ`. @@ -82,7 +85,6 @@ def coefficients(self): - """ Return the list coefficients of the ternary quadratic form. @@ -101,7 +103,6 @@ return self._a, self._b, self._c, self._r, self._s, self._t def coefficient(self,n): - """ Return the n-th coefficient of the ternary quadratic form, with 0<=n<=5. @@ -122,7 +123,6 @@ return self.coefficients()[n] def polynomial(self): - """ Return the polynomial associated to the ternary quadratic form. @@ -153,7 +153,7 @@ EXAMPLES:: sage: Q = TernaryQF([1, 1, 0, 2, -3, -1]) - sage: Q + sage: print Q._repr_() Ternary quadratic form with integer coefficients: [1 1 0] [2 -3 -1] @@ -219,7 +219,6 @@ def quadratic_form(self): - """ Return the object QuadraticForm with the same coefficients as Q over ZZ. @@ -241,7 +240,6 @@ return QuadraticForm(ZZ, 3, [self._a, self._t, self._s, self._b, self._r, self._c]) def matrix(self): - """ Return the Hessian matrix associated to the ternary quadratic form. That is, if Q is a ternary quadratic form, Q(x,y,z) = a*x^2 + b*y^2 + c*z^2 + r*y*z + s*x*z + t*x*y, @@ -267,7 +265,7 @@ sage: v = vector((1, 2, 3)) sage: Q(v) 28 - sage: (v*M*v.transpose())[0]//2 + sage: (v*M*v.column())[0]//2 28 """ @@ -276,7 +274,6 @@ return M def disc(self): - """ Return the discriminant of the ternary quadratic form, this is the determinant of the matrix divided by 2. @@ -293,7 +290,6 @@ return 4*self._a*self._b*self._c + self._r*self._s*self._t - self._a*self._r**2 - self._b*self._s**2 - self._c*self._t**2 def is_definite(self): - """ Determines if the ternary quadratic form is definite. @@ -337,7 +333,6 @@ return False def is_positive_definite(self): - """ Determines if the ternary quadratic form is positive definite. @@ -380,7 +375,6 @@ return False def is_negative_definite(self): - """ Determines if the ternary quadratic form is negatice definite. @@ -418,7 +412,6 @@ return False def __neg__(self): - """ Returns the ternary quadratic form with coefficients negatives of self. @@ -442,7 +435,6 @@ return TernaryQF([-a for a in self.coefficients()]) def is_primitive(self): - """ Determines if the ternary quadratic form is primitive, i.e. the greatest common divisor of the coefficients of the form is 1. @@ -463,7 +455,6 @@ return self.content() == 1 def primitive(self): - """ Returns the primitive version of the ternary quadratic form. @@ -491,7 +482,6 @@ return TernaryQF([a//g for a in l]) def scale_by_factor(self, k): - """ Scale the values of the ternary quadratic form by the number c, if c times the content of the ternary quadratic form is an integer it returns a ternary quadratic form, otherwise returns a quadratic form of dimension 3. @@ -534,7 +524,6 @@ raise TypeError, "Oops! " + k.__repr__() + " doesn't belongs to a Ring" def reciprocal(self): - """ Gives the reciprocal quadratic form associated to the given form. This is defined as the multiple of the primitive adjoint with the same content as the given form. @@ -558,7 +547,6 @@ return self.adjoint().primitive().scale_by_factor( self.content() ) def reciprocal_reduced(self): - """ Returns the reduced form of the reciprocal form of the given ternary quadratic form. @@ -581,7 +569,6 @@ return self.reciprocal().reduced_form_eisenstein(matrix = False) def divisor(self): - """ Returns the content of the adjoint form associated to the given form. @@ -602,7 +589,6 @@ return m def __eq__(self,right): - """ Determines if two ternary quadratic forms are equal. @@ -622,7 +608,6 @@ return self.coefficients() == right.coefficients() def adjoint(self): - """ Returns the adjoint form associated to the given ternary quadratic form. That is, the Hessian matrix of the adjoint form is twice the adjoint matrix of the Hessian matrix of the given form. @@ -648,7 +633,6 @@ return TernaryQF([A11, A22, A33, 2*A23, 2*A13, 2*A12]) def content(self): - """ Returns the greatest common divisor of the coefficients of the given ternary quadratic form. @@ -667,7 +651,6 @@ return gcd(self.coefficients()) def omega(self): - """ Returns the content of the adjoint of the primitive associated ternary quadratic form. @@ -685,7 +668,6 @@ return self.primitive().adjoint().content() def delta(self): - """ Returns the omega of the adjoint of the given ternary quadratic form, which is the same as the omega of the reciprocal form. @@ -708,7 +690,6 @@ return self.adjoint().omega() def level(self): - """ Returns the level of the ternary quadratic form, which is 4 times the discriminant divided by the divisor. @@ -725,7 +706,6 @@ return 4*self.disc()//self.divisor() def is_eisenstein_reduced(self): - """ Determines if the ternary quadratic form is Eisenstein reduced. That is, if we have a ternary quadratic form: @@ -804,7 +784,6 @@ return True def reduced_form_eisenstein(self, matrix=True): - """ Returns the eisenstein reduced form equivalent to the given positive ternary quadratic form, which is unique. @@ -843,7 +822,6 @@ return TernaryQF(v) def pseudorandom_primitive_zero_mod_p(self,p): - """ Returns a tuple of the form v = (a, b, 1) such that is a zero of the given ternary quadratic positive definite form modulo an odd prime p, where p doesn't divides the discriminant of the form. @@ -883,7 +861,6 @@ def find_zeros_mod_p(self, p): - """ Find the zeros of the given ternary quadratic positive definite form modulo a prime p, where p doesn't divides the discriminant of the form. @@ -917,7 +894,6 @@ def find_p_neighbor_from_vec(self, p, v, mat = False): - """ Finds the reduced equivalent of the p-neighbor of this ternary quadratic form associated to a given vector v satisfying: @@ -962,7 +938,6 @@ return TernaryQF(_find_p_neighbor_from_vec(self._a, self._b, self._c, self._r, self._s, self._t, p, v, mat)) def find_p_neighbors(self, p, mat = False): - """ Find a list with all the reduced equivalent of the p-neighbors of this ternary quadratic form, given by the zeros mod p of the form. See find_p_neighbor_from_vec for more information. @@ -993,16 +968,19 @@ def basic_lemma(self, p): - """ Finds a number represented by self and coprime to the prime p. + EXAMPLES:: + + sage: Q = TernaryQF([3, 3, 3, -2, 0, -1]) + sage: Q.basic_lemma(3) + 4 """ return _basic_lemma(self._a, self._b, self._c, self._r, self._s, self._t, p) def xi(self, p): - """ Return the value of the genus characters Xi_p... which may be missing one character. We allow -1 as a prime. @@ -1030,7 +1008,6 @@ return kronecker_symbol(self.basic_lemma(p), p) def xi_rec(self, p): - """ Returns Xi(p) for the reciprocal form. @@ -1052,7 +1029,6 @@ def symmetry(self, v): - """ Returns A the automorphism of the ternary quadratic form such that: :: @@ -1087,11 +1063,10 @@ """ - return identity_matrix(3) - v.transpose()*matrix(v)*self.matrix()/self(v) + return identity_matrix(3) - v.column()*matrix(v)*self.matrix()/self(v) def automorphism_symmetrys(self, A): - """ Given the automorphism A, returns two vectors v1, v2 if A is not the identity. Such that the product of the symmetries of the ternary quadratic form given by the two vectors is A. @@ -1136,7 +1111,6 @@ return [b1, b2] def automorphism_spin_norm(self,A): - """ Return the spin norm of the automorphism A. @@ -1161,10 +1135,9 @@ def _border(self,n): - """ Auxiliar function to find the automorphisms of a positive definite ternary quadratic form. - It return a boolen wether the n-condition is true. If Q = TernaryQF([a,b,c,r,s,t]), the conditions are: + It return a boolean whether the n-condition is true. If Q = TernaryQF([a,b,c,r,s,t]), the conditions are: :: 1- a = t, s = 2r. 2- a = s, t = 2r. @@ -1183,6 +1156,58 @@ 15- a = b, a + b + r + s + t = 0. 16- a = b, b = c, a + b + r + s + t = 0. + EXAMPLES:: + + sage: Q01 = TernaryQF([5, 5, 9, 2, 4, 5]) + sage: Q01._border(1) + True + sage: Q02 = TernaryQF([6, 7, 8, 2, 6, 4]) + sage: Q02._border(2) + True + sage: Q03 = TernaryQF([6, 9, 9, 9, 3, 6]) + sage: Q03._border(3) + True + sage: Q04 = TernaryQF([1, 2, 3, -1, 0, -1]) + sage: Q04._border(4) + True + sage: Q05 = TernaryQF([2, 3, 5, -1, -2, 0]) + sage: Q05._border(5) + True + sage: Q06 = TernaryQF([1, 5, 7, -5, 0, 0]) + sage: Q06._border(6) + True + sage: Q07 = TernaryQF([1, 1, 7, -1, -1, 0]) + sage: Q07._border(7) + True + sage: Q08 = TernaryQF([2, 2, 5, -1, -1, -1]) + sage: Q08._border(8) + True + sage: Q09 = TernaryQF([3, 8, 8, 6, 2, 2]) + sage: Q09._border(9) + True + sage: Q10 = TernaryQF([1, 3, 4, 0, 0, 0]) + sage: Q10._border(10) + True + sage: Q11 = TernaryQF([3, 5, 8, 0, -1, 0]) + sage: Q11._border(11) + True + sage: Q12 = TernaryQF([2, 6, 7, -5, 0, 0]) + sage: Q12._border(12) + True + sage: Q13 = TernaryQF([1, 1, 2, 1, 1, 1]) + sage: Q13._border(13) + True + sage: Q14 = TernaryQF([1, 3, 4, 3, 1, 1]) + sage: Q14._border(14) + True + sage: Q15 = TernaryQF([3, 3, 6, -3, -3, 0]) + sage: Q15._border(15) + True + sage: Q16 = TernaryQF([4, 4, 4, -2, -3, -3]) + sage: Q16._border(16) + True + + """ a, b, c, r, s, t = self.coefficients() @@ -1220,19 +1245,67 @@ return (a == b) and (b == c) and (a + b + r + s + t == 0) def _borders(self): - """ Return the borders that the ternary quadratic form meet. See: TernaryQF._border + EXAMPLES:: + + sage: Q01 = TernaryQF([5, 5, 9, 2, 4, 5]) + sage: Q01._borders() + (1,) + sage: Q02 = TernaryQF([6, 7, 8, 2, 6, 4]) + sage: Q02._borders() + (2,) + sage: Q03 = TernaryQF([6, 9, 9, 9, 3, 6]) + sage: Q03._borders() + (3,) + sage: Q04 = TernaryQF([1, 2, 3, -1, 0, -1]) + sage: Q04._borders() + (4,) + sage: Q05 = TernaryQF([2, 3, 5, -1, -2, 0]) + sage: Q05._borders() + (5,) + sage: Q06 = TernaryQF([1, 5, 7, -5, 0, 0]) + sage: Q06._borders() + (6, 12) + sage: Q07 = TernaryQF([1, 1, 7, -1, -1, 0]) + sage: Q07._borders() + (5, 6, 7, 8, 15) + sage: Q08 = TernaryQF([2, 2, 5, -1, -1, -1]) + sage: Q08._borders() + (8,) + sage: Q09 = TernaryQF([3, 8, 8, 6, 2, 2]) + sage: Q09._borders() + (9,) + sage: Q10 = TernaryQF([1, 3, 4, 0, 0, 0]) + sage: Q10._borders() + (10, 11, 12) + sage: Q11 = TernaryQF([3, 5, 8, 0, -1, 0]) + sage: Q11._borders() + (11,) + sage: Q12 = TernaryQF([2, 6, 7, -5, 0, 0]) + sage: Q12._borders() + (12,) + sage: Q13 = TernaryQF([1, 1, 2, 1, 1, 1]) + sage: Q13._borders() + (8, 13, 14) + sage: Q14 = TernaryQF([1, 3, 4, 3, 1, 1]) + sage: Q14._borders() + (14,) + sage: Q15 = TernaryQF([3, 3, 6, -3, -3, 0]) + sage: Q15._borders() + (5, 6, 7, 8, 15) + sage: Q16 = TernaryQF([4, 4, 4, -2, -3, -3]) + sage: Q16._borders() + (9, 15, 16) + """ return tuple(n for n in range(1,17) if self._border(n)) - def _automorphisms_reduced(self): - """ Return the coefficients of the matrices of the automorphisms of the reduced ternary quadratic form. @@ -1619,7 +1692,6 @@ def automorphisms(self): - """ Returns a list with the automorphisms of the definite ternary quadratic form. @@ -1683,7 +1755,6 @@ def _number_of_automorphisms_reduced(self): - """ Return the number of automorphisms of the reduced definite ternary quadratic form. @@ -1865,7 +1936,6 @@ def number_of_automorphisms(self): - """ Return the number of automorphisms of the definite ternary quadratic form. @@ -1908,7 +1978,6 @@ def find_all_ternary_qf_by_level_disc(N, d): - """ Find the coefficients of all the reduced ternary quadratic forms given it's discriminant d and level N. If N|4d and d|N^2, then it may be some forms with that discriminant and level. @@ -1947,7 +2016,6 @@ return map(TernaryQF, _find_all_ternary_qf_by_level_disc(N, d)) def find_a_ternary_qf_by_level_disc(N, d): - """ Find a reduced ternary quadratic form given it's discriminant d and level N. If N|4d and d|N^2, then it may be a form with that discriminant and level.