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@adeines
adeines committed Aug 22, 2016
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Showing 1 changed file with 176 additions and 20 deletions.
196 changes: 176 additions & 20 deletions src/sage/modules/free_module.py
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Original file line number Diff line number Diff line change
Expand Up @@ -166,10 +166,18 @@
from . import free_module_element
import sage.matrix.matrix_space
from sage.libs.pari.all import pari, pari_gen
import sage.misc.latex as latex

from sage.modules.module import Module
import sage.rings.finite_rings.finite_field_constructor as finite_field

import sage.misc.latex as latex

import sage.rings.number_field.order as nf_order
import sage.rings.commutative_ring as commutative_ring
import sage.rings.principal_ideal_domain as principal_ideal_domain
import sage.rings.field as field
import sage.rings.finite_rings.constructor as finite_field
import sage.rings.integral_domain as integral_domain
>>>>>>> Added a _pseudo_basis_matrix method to modules/free_module.py and
import sage.rings.ring as ring
import sage.rings.integer_ring
import sage.rings.rational_field
Expand Down Expand Up @@ -1187,7 +1195,7 @@ def basis_matrix(self, ring=None):
::
sage: M = FreeModule(GF(7),3).spang([[2,3,4],[1,1,1]]); M
sage: M = FreeModule(GF(7),3).span([[2,3,4],[1,1,1]]); M
Vector space of degree 3 and dimension 2 over Finite Field of size 7
Basis matrix:
[1 0 6]
Expand Down Expand Up @@ -1283,7 +1291,7 @@ def echelonized_basis_matrix(self):
...
NotImplementedError
"""
raise NoteImplementedError
raise NotImplementedError

def matrix(self):
"""
Expand Down Expand Up @@ -5363,6 +5371,34 @@ def __cmp__(self, other):
Free module of degree 1 and rank 1 over Integer Ring
Echelon basis matrix:
[1]
We now generalize this to class number one rings of integers.
::
sage: F.<a> = NumberField(x^2-x-1)
sage: ZF = F.ring_of_integers()
sage: A = matrix(ZF,[[1,0,0,0],[0,-1,0,0],[0,0,1,0],[0,0,0,1]])
sage: B = identity_matrix(ZF,4)
sage: VA = (ZF^4).span(A)
sage: VB = (ZF^4).span(B)
sage: VB == VA
True
Now we check an example with the same _pseudo_hermite_matrix matrices,
but different ideal lists::
sage: C = matrix(ZF,[[3,0,3,0],[0,3,6,9],[0,0,3,0],[12,0,0,3]])
sage: VC = (ZF^4).span(C)
sage: VC == VA
False
And finally and example where they have different _pseudo_hermite_matrix matrices::
sage: D = matrix(ZF,[[1+a,1,12*a,1],[1+a,1+a,1+a,1+a],[2*a+3,1+a,4*a+5,0],[1,0,1,1]])
sage: VD = (ZF^4).span(D)
sage: VA == VD
False
"""
if self is other:
Expand All @@ -5377,14 +5413,16 @@ def __cmp__(self, other):
if c: return c
# We use self.echelonized_basis_matrix() == other.echelonized_basis_matrix()
# with the matrix to avoid a circular reference.
from sage.rings.number_field.order import is_NumberFieldOrder
if is_NumberFieldOrder(self.base_ring()) and self.base_ring() != ZZ:
Mat,Ids = self._pseudo_hermite_matrix()
Omat,OIds = other._pseudo_hermite_matrix()
if Mat == Omat and Ids == OIds:
return True
else:
return False

if nf_order.is_NumberFieldOrder(self.base_ring()) and self.base_ring() != ZZ:
Mat,Ids = self._pseudo_hermite_matrix()
Omat,OIds = other._pseudo_hermite_matrix()
#if the ideals are the same, compare the matrices
if Ids == OIds:
return cmp(Mat,Omat)
#otherwise, compare the ideals.
else:
return cmp(Ids,OIds)
return cmp(self.echelonized_basis_matrix(), other.echelonized_basis_matrix())

def _denominator(self, B):
Expand Down Expand Up @@ -5498,20 +5536,138 @@ def _pseudo_hermite_matrix(self,ideal_list = None):
r"""
This returns the pseudo basis as a matrix in hermite form and a list of ideals.
It is here for now, but could be used more generally, e.g., for non-PIDs.
It is here for now, but could be used more generally, e.g., for non-PIDs, more specifically,
for non class number one rings of integers.
INPUT:
- `self` -- a free module over a PID, specifically a class number one ring of integers.
- `ideal_list` -- a list of ideals used to generate the pseudo basis.
OUTPUT:
A pseudo basis matrix in Hermite normal form and the list of pseodo basis ideals.
We start with an example of free modules that are equal, but due to extra units in the
number field ring of integers, Echelon form cannot tell them apart.
::
sage: F.<a> = NumberField(x^2-x-1)
sage: ZF = F.ring_of_integers()
sage: A = matrix(ZF,[[1,0,0,0],[0,-1,0,0],[0,0,1,0],[0,0,0,1]])
sage: B = identity_matrix(ZF,4)
sage: VA = (ZF^4).span(A)
sage: VA._pseudo_hermite_matrix()
(
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1], [Fractional ideal (1), Fractional ideal (1), Fractional ideal (1), Fractional ideal (1)]
)
sage: VA._pseudo_hermite_matrix([F.ideal(1)])
Traceback (most recent call last):
...
ValueError: gens and ideal_list must have the same lenghth
sage: VB = (ZF^4).span(B)
sage: VB._pseudo_hermite_matrix()
(
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1], [Fractional ideal (1), Fractional ideal (1), Fractional ideal (1), Fractional ideal (1)]
)
Now we check an example with non-trivial ideal list returned::
sage: C = matrix(ZF,[[3,0,3,0],[0,3,6,9],[0,0,3,0],[12,0,0,3]])
sage: VC = (ZF^4).span(C)
sage: VC._pseudo_hermite_matrix()
(
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1], [Fractional ideal (3), Fractional ideal (3), Fractional ideal (3), Fractional ideal (3)]
)
An example with a more interesting _pseudo_hermite_matrix matrix::
sage: D = matrix(ZF,[[1+a,1,12*a,1],[1+a,1+a,1+a,1+a],[2*a+3,1+a,4*a+5,0],[1,0,1,1]])
sage: VD = (ZF^4).span(D)
sage: VD._pseudo_hermite_matrix()
(
[ 1 0 0 618/1279*a + 551/1279]
[ 0 1 0 0]
[ 0 0 1 -50/1279*a - 533/1279]
[ 0 0 0 1], [Fractional ideal (1), Fractional ideal (1), Fractional ideal (1), Fractional ideal (-32*a + 15)]
)
An example with a non-trivial ideal_list input::
sage: IC = [F.fractional_ideal(1), F.fractional_ideal(2), F.fractional_ideal(2), F.fractional_ideal(2)]
sage: VA._pseudo_hermite_matrix(ideal_list = IC)
(
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1], [Fractional ideal (1), Fractional ideal (2), Fractional ideal (2), Fractional ideal (2)]
)
We can also handle ideal_lists over ZZ::
sage: M = identity_matrix(ZZ,10)
sage: ID = [ideal(ZZ(p)) for p in prime_range(30)]
sage: VZ = (ZZ^10).span(M)
sage: VZ._pseudo_hermite_matrix(ID)
[ 2 0 0 0 0 0 0 0 0 0]
[ 0 3 0 0 0 0 0 0 0 0]
[ 0 0 5 0 0 0 0 0 0 0]
[ 0 0 0 7 0 0 0 0 0 0]
[ 0 0 0 0 11 0 0 0 0 0]
[ 0 0 0 0 0 13 0 0 0 0]
[ 0 0 0 0 0 0 17 0 0 0]
[ 0 0 0 0 0 0 0 19 0 0]
[ 0 0 0 0 0 0 0 0 23 0]
[ 0 0 0 0 0 0 0 0 0 29]
sage: ID = [ideal(ZZ(p)) for p in prime_range(10)]
sage: VZ._pseudo_hermite_matrix(ID)
Traceback (most recent call last):
...
ValueError: gens and ideal_list must have the same lenghth
sage: M=matrix([[ZZ(i+j^2) for i in range(10)] for j in range(3,13)])
sage: VZ = (ZZ^10).span(M)
sage: VZ._pseudo_hermite_matrix()
[ 1 0 -1 -2 -3 -4 -5 -6 -7 -8]
[ 0 1 2 3 4 5 6 7 8 9]
"""
ZF = self._base_ring
assert is_NumberFieldOrder(ZF)
if ZF == ZZ:
return self._basis_matrix.hermite_form()
import sage.matrix.constructor
ZF = self._base
#do ZZ first
if ZF == ZZ and ideal_list == None:
return self.basis_matrix().hermite_form()
elif ZF == ZZ:
if len(self.gens()) != len(ideal_list):
raise ValueError("gens and ideal_list must have the same lenghth")
gens = [g*ids.gen() for g,ids in zip(self.gens(),ideal_list)]
M = sage.matrix.constructor.matrix(gens).hermite_form()
return M
#otherwise, it'd better be a number field maximal order.
#in this case we use Pari for pseudo bases.
assert nf_order.is_NumberFieldOrder(ZF)
F = ZF.number_field()
PF=pari(F)
BM = self._basis_matrix
BM = self.basis_matrix()
if ideal_list == None:
ideal_list = [F.ideal(1) for id in range(BM.nrows())]
ideal_list = [F.ideal(1) for id in range(len(self.gens()))]
elif len(self.gens()) != len(ideal_list):
raise ValueError("gens and ideal_list must have the same lenghth")
pari_ideal_list = [pari(id) for id in ideal_list]
HM,HI = PF.nfhnf([pari(BM),pari_ideal_list])
M = matrix([[F(HM[i][j]) for i in range(len(HM))] for j in range(len(HM[0]))])

M = sage.matrix.constructor.matrix([[F(HM[i][j]) for i in range(len(HM))] for j in range(len(HM[0]))])
I = [F.ideal(id) for id in HI]
return M, I

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