Coverage for local/lib/python2.7/site-packages/sage/schemes/plane_conics/con_rational_function_field.py : 87%

# -*- coding: utf-8 -*- 

r""" 

Projective plane conics over a rational function field 

The class :class:`ProjectiveConic_rational_function_field` represents a 

projective plane conic over a rational function field `F(t)`, where `F` 

is any field. Instances can be created using :func:`Conic`. 

AUTHORS: 

- Lennart Ackermans (2016-02-07): initial version 

EXAMPLES: 

Create a conic:: 

sage: K = FractionField(PolynomialRing(QQ, 't')) 

sage: P.<X, Y, Z> = K[] 

sage: Conic(X^2 + Y^2 - Z^2) 

Projective Conic Curve over Fraction Field of Univariate 

Polynomial Ring in t over Rational Field defined by 

X^2 + Y^2 - Z^2 

Points can be found using :meth:`has_rational_point`:: 

sage: K.<t> = FractionField(QQ['t']) 

sage: C = Conic([1,-t,t]) 

sage: C.has_rational_point(point = True) 

(True, (0 : 1 : 1)) 

""" 

#***************************************************************************** 

# Copyright (C) 2016 Lennart Ackermans 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import division 

from __future__ import absolute_import 

from sage.rings.all import PolynomialRing 

from sage.matrix.constructor import diagonal_matrix, matrix, block_matrix 

from sage.schemes.plane_conics.con_field import ProjectiveConic_field 

from sage.arith.all import lcm, gcd 

from sage.modules.free_module_element import vector 

from sage.rings.fraction_field import is_FractionField 

class ProjectiveConic_rational_function_field(ProjectiveConic_field): 

r""" 

Create a projective plane conic curve over a rational function field 

`F(t)`, where `F` is any field. 

The algorithms used in this class come mostly from [HC2006]_. 

EXAMPLES:: 

sage: K = FractionField(PolynomialRing(QQ, 't')) 

sage: P.<X, Y, Z> = K[] 

sage: Conic(X^2 + Y^2 - Z^2) 

Projective Conic Curve over Fraction Field of Univariate 

Polynomial Ring in t over Rational Field defined by 

X^2 + Y^2 - Z^2 

TESTS:: 

sage: K = FractionField(PolynomialRing(QQ, 't')) 

sage: Conic([K(1), 1, -1])._test_pickling() 

REFERENCES: 

.. [HC2006] Mark van Hoeij and John Cremona, Solving Conics over 

function fields. J. Théor. Nombres Bordeaux, 2006. 

.. [ACKERMANS2016] Lennart Ackermans, Oplosbaarheid van Kegelsneden. 

http://www.math.leidenuniv.nl/nl/theses/Bachelor/. 

""" 

def __init__(self, A, f): 

r""" 

See ``Conic`` for full documentation. 

EXAMPLES:: 

sage: c = Conic([1, 1, 1]); c 

Projective Conic Curve over Rational Field defined by 

x^2 + y^2 + z^2 

""" 

ProjectiveConic_field.__init__(self, A, f) 

def has_rational_point(self, point = False, algorithm = 'default', 

read_cache = True): 

r""" 

Returns True if and only if the conic ``self`` 

has a point over its base field `F(t)`, which is a field of rational 

functions. 

If ``point`` is True, then returns a second output, which is 

a rational point if one exists. 

Points are cached whenever they are found. Cached information 

is used if and only if ``read_cache`` is True. 

The default algorithm does not (yet) work for all base fields `F`. 

In particular, sage is required to have: 

* an algorithm for finding the square root of elements in finite 

extensions of `F`; 

* a factorization and gcd algorithm for `F[t]`; 

* an algorithm for solving conics over `F`. 

ALGORITHM: 

The parameter ``algorithm`` specifies the algorithm 

to be used: 

* ``'default'`` -- use a native Sage implementation, based on the 

algorithm Conic in [HC2006]_. 

* ``'magma'`` (requires Magma to be installed) -- 

delegates the task to the Magma computer algebra 

system. 

EXAMPLES: 

We can find points for function fields over (extensions of) `\QQ` 

and finite fields:: 

sage: K.<t> = FractionField(PolynomialRing(QQ, 't')) 

sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18]) 

sage: C.has_rational_point(point=True) 

(True, (-3 : (t + 1)/t : 1)) 

sage: R.<t> = FiniteField(23)[] 

sage: C = Conic([2, t^2+1, t^2+5]) 

sage: C.has_rational_point() 

True 

sage: C.has_rational_point(point=True) 

(True, (5*t : 8 : 1)) 

sage: F.<i> = QuadraticField(-1) 

sage: R.<t> = F[] 

sage: C = Conic([1,i*t,-t^2+4]) 

sage: C.has_rational_point(point = True) 

(True, (-t - 2*i : -2*i : 1)) 

It works on non-diagonal conics as well:: 

sage: K.<t> = QQ[] 

sage: C = Conic([4, -4, 8, 1, -4, t + 4]) 

sage: C.has_rational_point(point=True) 

(True, (1/2 : 1 : 0)) 

If no point exists output still depends on the argument ``point``:: 

sage: K.<t> = QQ[] 

sage: C = Conic(K, [t^2, (t-1), -2*(t-1)]) 

sage: C.has_rational_point() 

False 

sage: C.has_rational_point(point=True) 

(False, None) 

Due to limitations in Sage of algorithms we depend on, it is not 

yet possible to find points on conics over multivariate function fields 

(see the requirements above):: 

sage: F.<t1> = FractionField(QQ['t1']) 

sage: K.<t2> = FractionField(F['t2']) 

sage: a = K(1) 

sage: b = 2*t2^2+2*t1*t2-t1^2 

sage: c = -3*t2^4-4*t1*t2^3+8*t1^2*t2^2+16*t1^3-t2-48*t1^4 

sage: C = Conic([a,b,c]) 

sage: C.has_rational_point() 

... 

Traceback (most recent call last): 

... 

NotImplementedError: is_square() not implemented for elements of 

Univariate Quotient Polynomial Ring in tbar over Fraction Field 

of Univariate Polynomial Ring in t1 over Rational Field with 

modulus tbar^2 + t1*tbar - 1/2*t1^2 

In some cases, the algorithm requires us to be 

able to solve conics over `F`. In particular, the following does not 

work:: 

sage: P.<u> = QQ[] 

sage: E = P.fraction_field() 

sage: Q.<Y> = E[] 

sage: F.<v> = E.extension(Y^2 - u^3 - 1) 

sage: R.<t> = F[] 

sage: K = R.fraction_field() 

sage: C = Conic(K, [u, v, 1]) 

sage: C.has_rational_point() 

... 

Traceback (most recent call last): 

... 

NotImplementedError: has_rational_point not implemented for conics 

over base field Univariate Quotient Polynomial Ring in v over 

Fraction Field of Univariate Polynomial Ring in u over Rational 

Field with modulus v^2 - u^3 - 1 

``has_rational_point`` fails for some conics over function fields 

over finite fields, due to :trac:`20003`:: 

sage: K.<t> = PolynomialRing(GF(7)) 

sage: C = Conic([5*t^2+4, t^2+3*t+3, 6*t^2+3*t+2, 5*t^2+5, 4*t+3, 4*t^2+t+5]) 

sage: C.has_rational_point() 

... 

Traceback (most recent call last): 

... 

TypeError: self (=Scheme morphism: 

From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + ((6*t^3 + 3*t^2 + 5*t + 5)/(t + 3))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z^2 

To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + (t^2 + 3*t + 3)*x*y + (5*t^2 + 5)*y^2 + (6*t^2 + 3*t + 2)*x*z + (4*t + 3)*y*z + (4*t^2 + t + 5)*z^2 

Defn: Defined on coordinates by sending (x : y : z) to 

(x + ((2*t + 5)/(t + 3))*y + ((3*t^4 + 2*t^3 + 5*t^2 + 5*t + 3)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z : y + ((6*t^3 + 6*t^2 + 3*t + 6)/(t^3 + 4*t^2 + 2*t + 2))*z : z)) domain must equal right (=Scheme morphism: 

From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^3 + 6*t^2 + 3*t + 3)*x^2 + (t + 4)*y^2 + (6*t^7 + 2*t^5 + t^4 + 2*t^3 + 3*t^2 + 6*t + 6)*z^2 

To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5/(t^3 + 4*t^2 + 2*t + 2))*x^2 + (1/(t^3 + 3*t^2 + 5*t + 1))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^9 + 5*t^8 + t^7 + 6*t^6 + 3*t^5 + 4*t^3 + t^2 + 5*t + 3))*z^2 

Defn: Defined on coordinates by sending (x : y : z) to 

((t^3 + 4*t^2 + 2*t + 2)*x : (t^2 + 5)*y : (t^5 + 4*t^4 + t^2 + 3*t + 3)*z)) codomain 

TESTS:: 

sage: K.<t> = FractionField(PolynomialRing(QQ, 't')) 

sage: a = (2*t^2 - 3/2*t + 1)/(37/3*t^2 + t - 1/4) 

sage: b = (1/2*t^2 + 1/3)/(-73*t^2 - 2*t + 11/4) 

sage: c = (6934/3*t^6 + 8798/3*t^5 - 947/18*t^4 + 3949/9*t^3 + 20983/18*t^2 + 28/3*t - 131/3)/(-2701/3*t^4 - 293/3*t^3 + 301/6*t^2 + 13/4*t - 11/16) 

sage: C = Conic([a,b,c]) 

sage: C.has_rational_point(point=True) 

(True, (4*t + 4 : 2*t + 2 : 1)) 

A long time test:: 

sage: K.<t> = FractionField(PolynomialRing(QQ, 't')) 

sage: a = (-1/3*t^6 - 14*t^5 - 1/4*t^4 + 7/2*t^2 - 1/2*t - 1)/(24/5*t^6 - t^5 - 1/4*t^4 + t^3 - 3*t^2 + 8/5*t + 5) 

sage: b = (-3*t^3 + 8*t + 1/2)/(-1/3*t^3 + 3/2*t^2 + 1/12*t + 1/2) 

sage: c = (1232009/225*t^25 - 1015925057/8100*t^24 + 1035477411553/1458000*t^23 + 7901338091/30375*t^22 - 1421379260447/729000*t^21 + 266121260843/972000*t^20 + 80808723191/486000*t^19 - 516656082523/972000*t^18 + 21521589529/40500*t^17 + 4654758997/21600*t^16 - 20064038625227/9720000*t^15 - 173054270347/324000*t^14 + 536200870559/540000*t^13 - 12710739349/50625*t^12 - 197968226971/135000*t^11 - 134122025657/810000*t^10 + 22685316301/120000*t^9 - 2230847689/21600*t^8 - 70624099679/270000*t^7 - 4298763061/270000*t^6 - 41239/216000*t^5 - 13523/36000*t^4 + 493/36000*t^3 + 83/2400*t^2 + 1/300*t + 1/200)/(-27378/125*t^17 + 504387/500*t^16 - 97911/2000*t^15 + 1023531/4000*t^14 + 1874841/8000*t^13 + 865381/12000*t^12 + 15287/375*t^11 + 6039821/6000*t^10 + 599437/1500*t^9 + 18659/250*t^8 + 1218059/6000*t^7 + 2025127/3000*t^6 + 1222759/6000*t^5 + 38573/200*t^4 + 8323/125*t^3 + 15453/125*t^2 + 17031/500*t + 441/10) 

sage: C = Conic([a,b,c]) 

sage: C.has_rational_point(point = True) # long time (4 seconds) 

(True, 

((-2/117*t^8 + 304/1053*t^7 + 40/117*t^6 - 1/27*t^5 - 110/351*t^4 - 2/195*t^3 + 11/351*t^2 + 1/117)/(t^4 + 2/39*t^3 + 4/117*t^2 + 2/39*t + 14/39) : -5/3*t^4 + 19*t^3 : 1)) 

""" 

from .constructor import Conic 

if read_cache: 

if self._rational_point is not None: 

return (True, self._rational_point) if point else True 

if algorithm != 'default': 

return ProjectiveConic_field.has_rational_point(self, point, 

algorithm, read_cache) 

# Default algorithm 

if self.base_ring().characteristic() == 2: 

raise NotImplementedError("has_rational_point not implemented \ 

for function field of characteristic 2.") 

new_conic, transformation, inverse = self.diagonalization() 

coeff = new_conic.coefficients() 

if coeff[0] == 0: 

return (True, transformation([1,0,0])) if point else True 

elif coeff[3] == 0: 

return (True, transformation([0,1,0])) if point else True 

elif coeff[5] == 0: 

return (True, transformation([0,0,1])) if point else True 

# We save the coefficients of the reduced form in coeff 

# A zero of the reduced conic can be multiplied by multipliers 

# to get a zero of the old conic 

(coeff, multipliers) = new_conic._reduce_conic() 

new_conic = Conic(coeff) 

transformation = transformation \ 

* new_conic.hom(diagonal_matrix(multipliers)) 

if coeff[0].degree() % 2 == coeff[1].degree() % 2 and \ 

coeff[1].degree() % 2 == coeff[2].degree() % 2: 

case = 0 

else: 

case = 1 

t, = self.base_ring().base().gens() # t in F[t] 

supp = [] 

roots = [[], [], []] 

remove = None 

# loop through the coefficients and find a root of f_i (as in 

# [HC2006]) modulo each element in the coefficients' support 

for i in (0,1,2): 

supp.append(list(coeff[i].factor())) 

for p in supp[i]: 

if p[1] != 1: 

raise ValueError("Expected factor of exponent 1.") 

# Convert to monic factor 

x = p[0]/list(p[0])[-1] 

N = p[0].base_ring().extension(x, 'tbar') 

R = PolynomialRing(N, 'u') 

u, = R.gens() 

# If p[0] has degree 1, sage might forget the "defining 

# polynomial" of N, so we define our own modulo operation 

if p[0].degree() == 1: 

mod = t.parent().hom([-x[0]]) 

else: 

mod = N 

if i == 0: 

x = -mod(coeff[2])/mod(coeff[1]) 

elif i == 1: 

x = -mod(coeff[0])/mod(coeff[2]) 

else: 

x = -mod(coeff[1])/mod(coeff[0]) 

if x.is_square(): 

root = N(x.sqrt()) 

else: 

return (False, None) if point else False 

# if case == 0 and p[0] has degree 1, we switch to case 

# 1 and remove this factor out of the support. In [HC2006] 

# this is done later, in FindPoint. 

if case == 0 and p[0].degree() == 1: 

case = 1 

# remove later so the loop iterator stays in place. 

remove = (i,p) 

else: 

roots[i].append(root) 

if remove: 

supp[remove[0]].remove(remove[1]) 

supp = [[p[0] for p in supp[i]] for i in (0,1,2)] 

if case == 0: 

# Find a solution of (5) in [HC2006] 

leading_conic = Conic(self.base_ring().base_ring(), 

[coeff[0].leading_coefficient(), 

coeff[1].leading_coefficient(), 

coeff[2].leading_coefficient()]) 

has_point = leading_conic.has_rational_point(True) 

if has_point[0]: 

if point: 

pt = new_conic.find_point(supp, roots, case, 

has_point[1]) 

else: 

pt = True 

return (True, transformation(pt)) if point else True 

else: 

return (False, None) if point else False 

# case == 1: 

if point: 

pt = new_conic.find_point(supp, roots, case) 

else: 

pt = True 

return (True, transformation(pt)) if point else True 

def _reduce_conic(self): 

r""" 

Return the reduced form of the conic, i.e. a conic with base field 

`K=F(t)` and coefficients `a,b,c` such that `a,b,c \in F[t]`, 

`\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1` and `abc` is square-free. 

Assumes `self` is in diagonal form. 

OUTPUT: 

A tuple (coefficients, multipliers), the coefficients of the conic 

in reduced form and multipliers `\lambda, \mu, \nu \in F(t)^*` such 

that `(x,y,z) \in F(t)` is a solution of the reduced conic if and only 

if `(\lambda x, \mu y, \nu z)` is a solution of `self`. 

ALGORITHM: 

The algorithm used is the algorithm ReduceConic in [HC2006]_. 

EXAMPLES:: 

sage: K.<t> = FractionField(PolynomialRing(QQ, 't')) 

sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18]) 

sage: C._reduce_conic() 

([t^2 - 2, 2*t, -2*t^3 - 13*t^2 - 2*t + 18], [t, 1, t]) 

""" 

# start with removing fractions 

coeff = [self.coefficients()[0], self.coefficients()[3], 

self.coefficients()[5]] 

coeff = lcm(lcm(coeff[0].denominator(), coeff[1].denominator()), 

coeff[2].denominator()) * vector(coeff) 

# go to base ring of fraction field 

coeff = [self.base().base()(x) for x in coeff] 

coeff = vector(coeff) / gcd(coeff) 

# remove common divisors 

labda = mu = nu = 1 

g1 = g2 = g3 = 0 

ca, cb, cc = coeff 

while g1 != 1 or g2 != 1 or g3 != 1: 

g1 = gcd(ca,cb); ca = ca/g1; cb = cb/g1; cc = cc*g1; nu = g1*nu 

g2 = gcd(ca,cc); ca = ca/g2; cc = cc/g2; cb = cb*g2; mu = g2*mu 

g3 = gcd(cb,cc); cb = cb/g3; cc = cc/g3; ca = ca*g3; 

labda = g3*labda 

coeff = [ca, cb, cc] 

multipliers = [labda, mu, nu] 

# remove squares 

for i, x in enumerate(coeff): 

if is_FractionField(x.parent()): 

# go to base ring of fraction field 

x = self.base().base()(x) 

try: 

decom = x.squarefree_decomposition() 

except (NotImplementedError, AttributeError): 

decom = x.factor() 

x = decom.unit() 

x2 = 1 

for factor in decom: 

if factor[1] > 1: 

if factor[1] % 2 == 0: 

x2 *= factor[0] ** (factor[1] // 2) 

else: 

x *= factor[0] 

x2 *= factor[0] ** ((factor[1] - 1) // 2) 

else: 

x *= factor[0] 

for j, y in enumerate(multipliers): 

if j != i: 

multipliers[j] = y * x2 

coeff[i] = self.base_ring().base().coerce(x); 

return (coeff, multipliers) 

def find_point(self, supports, roots, case, solution = 0): 

r""" 

Given a solubility certificate like in [HC2006]_, find a point on 

``self``. Assumes ``self`` is in reduced form (see [HC2006]_ for a 

definition). 

If you don't have a solubility certificate and just want to find a 

point, use the function :meth:`has_rational_point` instead. 

INPUT: 

- ``self`` -- conic in reduced form. 

- ``supports`` -- 3-tuple where ``supports[i]`` is a list of all monic 

irreducible `p \in F[t]` that divide the `i`'th of the 3 coefficients. 

- ``roots`` -- 3-tuple containing lists of roots of all elements of 

``supports[i]``, in the same order. 

- ``case`` -- 1 or 0, as in [HC2006]_. 

- ``solution`` -- (default: 0) a solution of (5) in [HC2006]_, if 

case = 0, 0 otherwise. 

OUTPUT: 

A point `(x,y,z) \in F(t)` of ``self``. Output is undefined when the 

input solubility certificate is incorrect. 

ALGORITHM: 

The algorithm used is the algorithm FindPoint in [HC2006]_, with 

a simplification from [ACKERMANS2016]_. 

EXAMPLES:: 

sage: K.<t> = FractionField(QQ['t']) 

sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18]) 

sage: C.has_rational_point(point=True) # indirect test 

(True, (-3 : (t + 1)/t : 1)) 

Different solubility certificates give different points:: 

sage: K.<t> = PolynomialRing(QQ, 't') 

sage: C = Conic(K, [t^2-2, 2*t, -2*t^3-13*t^2-2*t+18]) 

sage: supp = [[t^2 - 2], [t], [t^3 + 13/2*t^2 + t - 9]] 

sage: tbar1 = QQ.extension(supp[0][0], 'tbar').gens()[0] 

sage: tbar2 = QQ.extension(supp[1][0], 'tbar').gens()[0] 

sage: tbar3 = QQ.extension(supp[2][0], 'tbar').gens()[0] 

sage: roots = [[tbar1 + 1], [1/3*tbar2^0], [2/3*tbar3^2 + 11/3*tbar3 - 3]] 

sage: C.find_point(supp, roots, 1) 

(3 : t + 1 : 1) 

sage: roots = [[-tbar1 - 1], [-1/3*tbar2^0], [-2/3*tbar3^2 - 11/3*tbar3 + 3]] 

sage: C.find_point(supp, roots, 1) 

(3 : -t - 1 : 1) 

""" 

Ft = self.base().base() 

F = Ft.base() 

t, = Ft.gens() 

coefficients = [Ft(self.coefficients()[0]), Ft(self.coefficients()[3]), 

Ft(self.coefficients()[5])] 

deg = [coefficients[0].degree(), coefficients[1].degree(), 

coefficients[2].degree()] 

# definitions as in [HC2006] and [ACKERMANS2016] 

A = ((deg[1] + deg[2]) / 2).ceil() - case 

B = ((deg[2] + deg[0]) / 2).ceil() - case 

C = ((deg[0] + deg[1]) / 2).ceil() - case 

# For all roots as calculated by has_rational_point(), we create 

# a system of linear equations. As in [ACKERMANS2016], we do this 

# by calculating the matrices for all phi_p, with basis consisting 

# of monomials of x, y and z in the space V of potential solutions: 

# t^0, ..., t^A, t^0, ..., t^B and t^0, ..., t^C. 

phi = [] 

for (i, p) in enumerate(supports[0]): 

# lift to F[t] and map to R, with R as defined above 

if roots[0][i].parent().is_finite(): 

root = roots[0][i].polynomial() 

else: 

root = roots[0][i].lift() 

alpha = root.parent().hom([t])(root) 

d = p.degree() 

# Calculate y - alpha*z mod p for all basis vectors 

phi_p = [[] for i in range(A+B+C+4)] 

phi_p[0:A+1] = [vector(F, d)] * (A+1) 

phi_p[A+1] = vector(F, d, {0: F(1)}) 

lastpoly = F(1) 

for n in range(B): 

lastpoly = (lastpoly * t) % p 

phi_p[A+2+n] = vector(F, d, lastpoly.dict()) 

lastpoly = -alpha % p 

phi_p[A+B+2] = vector(F, d, lastpoly.dict()) 

for n in range(C): 

lastpoly = (lastpoly * t) % p 

phi_p[A+B+3+n] = vector(F, d, lastpoly.dict()) 

phi_p[A+B+C+3] = vector(F, d) 

phi.append(matrix(phi_p).transpose()) 

for (i, p) in enumerate(supports[1]): 

if roots[1][i].parent().is_finite(): 

root = roots[1][i].polynomial() 

else: 

root = roots[1][i].lift() 

alpha = root.parent().hom([t])(root) 

d = p.degree() 

# Calculate z - alpha*x mod p for all basis vectors 

phi_p = [[] for i in range(A+B+C+4)] 

phi_p[A+1:A+B+2] = [vector(F, d)] * (B+1) 

phi_p[A+B+2] = vector(F, d, {0: F(1)}) 

lastpoly = F(1) 

for n in range(C): 

lastpoly = (lastpoly * t) % p 

phi_p[A+B+3+n] = vector(F, d, lastpoly.dict()) 

lastpoly = -alpha % p 

phi_p[0] = vector(F, d, lastpoly.dict()) 

for n in range(A): 

lastpoly = (lastpoly * t) % p 

phi_p[1+n] = vector(F, d, lastpoly.dict()) 

phi_p[A+B+C+3] = vector(F, d) 

phi.append(matrix(phi_p).transpose()) 

for (i, p) in enumerate(supports[2]): 

if roots[2][i].parent().is_finite(): 

root = roots[2][i].polynomial() 

else: 

root = roots[2][i].lift() 

alpha = root.parent().hom([t])(root) 

d = p.degree() 

# Calculate x - alpha*y mod p for all basis vectors 

phi_p = [[] for i in range(A+B+C+4)] 

phi_p[A+B+2:A+B+C+3] = [vector(F, d)] * (C+1) 

phi_p[0] = vector(F, d, {0: F(1)}) 

lastpoly = F(1) 

for n in range(A): 

lastpoly = (lastpoly * t) % p 

phi_p[1+n] = vector(F, d, lastpoly.dict()) 

lastpoly = -alpha % p 

phi_p[A+1] = vector(F, d, lastpoly.dict()) 

for n in range(B): 

lastpoly = (lastpoly * t) % p 

phi_p[A+2+n] = vector(F, d, lastpoly.dict()) 

phi_p[A+B+C+3] = vector(F, d) 

phi.append(matrix(phi_p).transpose()) 

if case == 0: 

# We need three more equations 

lx = Ft(solution[0]).leading_coefficient() 

ly = Ft(solution[1]).leading_coefficient() 

lz = Ft(solution[2]).leading_coefficient() 

phi.append(matrix([vector(F, A+B+C+4, {A:1, A+B+C+3:-lx}), 

vector(F, A+B+C+4, {A+B+1:1, A+B+C+3:-ly}), 

vector(F, A+B+C+4, {A+B+C+2: 1, A+B+C+3:-lz})])) 

# Create the final matrix which we will solve 

M = block_matrix(phi, ncols = 1, subdivide = False) 

solution_space = M.right_kernel() 

for v in solution_space.basis(): 

if v[:A+B+C+3] != 0: # we don't want to return a trivial solution 

X = Ft(list(v[:A+1])) 

Y = Ft(list(v[A+1:A+B+2])) 

Z = Ft(list(v[A+B+2:A+B+C+3])) 

return self.point([X,Y,Z]) 

raise RuntimeError("No solution has been found: possibly incorrect\ 

solubility certificate.")