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Laurent polynomials, Fitting ideals and characteristic varieties #36368

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27c60d7
add is_noetherian and divides, for univariate
enriqueartal Sep 20, 2023
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.
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divides
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avoid non integral domains
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typo in __contains__
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reduction of matrices of Laurent polynomials
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97bd76c
import identity
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doctests
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homogeneize __reduce__ for uni- and multi-variate Laurent polynomials
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style
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change __reduce__ again to avoid further errors
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Merge branch 'laurent' of github.com:enriqueartal/sage into laurent
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Merge branch 'sagemath:develop' into laurent
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905d042
class of matrices of Laurent mpolynomials
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use fitting_ideals for characteristic varieties
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68a94e5
adding xgcd, etc, for laurent polynomials in one variable
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8da8781
cut characteristic varieties when the total ideal is reached
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1153822
trailing spaces
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redefine gens_reduced and more doctests
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sum of ideals
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change hint definition
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undo format changes for finitely_presented.py
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Merge branch 'sagemath:develop' into laurent
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after first review
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changes in divides
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trick to solve the hint issue
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Merge branch 'sagemath:develop' into laurent
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descripion of Laurent matrix class
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tests polynomial ideal
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doctests for change_ring in ideal_py and __contains_ in laurent_polyn…
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change in monomial_reduction
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some cases missing in characteristic varieties
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136 changes: 94 additions & 42 deletions src/sage/groups/finitely_presented.py
Original file line number Diff line number Diff line change
Expand Up @@ -145,6 +145,7 @@
from sage.sets.set import Set
from sage.structure.unique_representation import UniqueRepresentation


class GroupMorphismWithGensImages(SetMorphism):
r"""
Class used for morphisms from finitely presented groups to
Expand Down Expand Up @@ -1702,12 +1703,18 @@ def abelian_alexander_matrix(self, ring=QQ, simplified=True):
sage: g = FreeGroup(1) / []
sage: g.abelian_alexander_matrix()
([], [])
sage: g.abelian_alexander_matrix()[0].base_ring()
Univariate Laurent Polynomial Ring in f1 over Rational Field
sage: g = FreeGroup(0) / []
sage: g.abelian_alexander_matrix()
([], [])
sage: A, ideal = g.abelian_alexander_matrix(); A
[]
sage: A.base_ring()
Rational Field
"""
ab, R, ideal, images = self.abelianization_to_algebra(ring=ring)
A = self.alexander_matrix(im_gens=images)
if A.base_ring() != R:
A = A.change_ring(R)
if simplified:
n, m = A.dimensions()
if n == 0 or m == 0:
Expand Down Expand Up @@ -1761,66 +1768,111 @@ def characteristic_varieties(self, ring=QQ, matrix_ideal=None, groebner=False):

OUTPUT:

If ``groebner`` is ``False`` a list of ideals defining the characteristic varieties.
If it is ``True``, a list of lists for Gröbner bases for each ideal.
A dictionary with keys the indices of the varieties. If ``groebner`` is ``False``
the values are the ideals defining the characteristic varieties.
If it is ``True``, lists for Gröbner bases for the ideal of each irreducible
component, stopping when the first time a characteristic variety is empty.

EXAMPLES::

sage: L = [2*(i, j) + 2* (-i, -j) for i, j in ((1, 2), (2, 3), (3, 1))]
sage: G = FreeGroup(3) / L
sage: G.characteristic_varieties(groebner=True)
[[(f1 - 1, f2 - 1, f3 - 1),
(f1 + 1, f2 - 1, f3 - 1),
(f1 - 1, f2 - 1, f3 + 1),
(f3^2 + 1, f1 - f3, f2 - f3),
(f1 - 1, f2 + 1, f3 - 1)],
[(f1 - 1, f2 - 1, f3 - 1),
(f1*f3 + 1, f2 - 1),
(f1*f2 + 1, f3 - 1),
(f2*f3 + 1, f1 - 1),
(f2*f3 + 1, f1 - f2),
(f2*f3 + 1, f1 - f3),
(f1*f3 + 1, f2 - f3)]]
{0: [(0,)],
1: [(f1 - 1, f2 - 1, f3 - 1), (f1*f3 + 1, f2 - 1), (f1*f2 + 1, f3 - 1), (f2*f3 + 1, f1 - 1),
(f2*f3 + 1, f1 - f2), (f2*f3 + 1, f1 - f3), (f1*f3 + 1, f2 - f3)],
2: [(f1 - 1, f2 - 1, f3 - 1), (f1 + 1, f2 - 1, f3 - 1), (f1 - 1, f2 - 1, f3 + 1),
(f3^2 + 1, f1 - f3, f2 - f3), (f1 - 1, f2 + 1, f3 - 1)],
3: [(f1 - 1, f2 - 1, f3 - 1)],
4: []}
sage: G = FreeGroup(2)/[2*(1,2,-1,-2)]
sage: G.characteristic_varieties()
[Ideal (-2*f2 + 2, 2*f1 - 2) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field]
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
1: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field}
sage: G.characteristic_varieties(ring=ZZ)
[Ideal (-2*f2 + 2, 2*f1 - 2) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring]
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
1: Ideal (2*f2 - 2, 2*f1 - 2) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring}
sage: G = FreeGroup(2)/[(1,2,1,-2,-1,-2)]
sage: G.characteristic_varieties()
[Ideal (1 - f2 + f2^2, -1 + f2 - f2^2) of Univariate Laurent Polynomial Ring in f2 over Rational Field]
{0: Ideal (0) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
1: Ideal (-1 + 2*f2 - 2*f2^2 + f2^3) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
2: Ideal (1) of Univariate Laurent Polynomial Ring in f2 over Rational Field}
sage: G.characteristic_varieties(groebner=True)
{0: [0], 1: [-1 + f2, 1 - f2 + f2^2], 2: []}
sage: G = FreeGroup(2)/[3 * (1, ), 2 * (2, )]
sage: G.characteristic_varieties(groebner=True)
[[1 - f2 + f2^2]]
{0: [-1 + F1, 1 + F1, 1 - F1 + F1^2, 1 + F1 + F1^2], 1: [1 - F1 + F1^2], 2: []}
sage: G = FreeGroup(2)/[2 * (2, )]
sage: G.characteristic_varieties(groebner=True)
{0: [(f1 + 1,), (f1 - 1,)], 1: [(f1 + 1,), (f1 - 1, f2 - 1)], 2: []}
sage: G = (FreeGroup(0) / [])
sage: G.characteristic_varieties()
{0: Principal ideal (0) of Rational Field,
1: Principal ideal (1) of Rational Field}
sage: G.characteristic_varieties(groebner=True)
{0: [(0,)], 1: [(1,)]}
"""
A, ideal = self.abelian_alexander_matrix(ring=ring, simplified=True)
if self.ngens() == 0:
if groebner:
return {j: [(ring(j),)] for j in (0, 1)}
return {j: ring.ideal(j) for j in (0, 1)}
A, rels = self.abelian_alexander_matrix(ring=ring, simplified=True)
R = A.base_ring()
res = []
eval_1 = {x: ring(1) for x in R.gens()}
A_scalar = A.apply_map(lambda p: p.subs(eval_1))
n = A.ncols()
n1 = n - A_scalar.rank()
ideal_1 = R.ideal([x - 1 for x in R.gens()])
S = R.polynomial_ring()
ideal = [S(elt) for elt in ideal]
for j in range(1, A.ncols()):
L = [p.monomial_reduction()[0] for p in A.minors(j)]
J = R.ideal(L + ideal)
res.append(J)
if not groebner or not R.base_ring().is_field():
K = R.base_ring()
id_rels = R.ideal(rels)
res = dict()
bound = n + 1
for j in range(bound + 1):
J = id_rels + A.fitting_ideal(j)
# J = R.ideal(id_rels.gens() + A.fitting_ideal(j).gens())
if j <= n1:
J1 = K.ideal([K(p.subs(eval_1)) for p in J.gens()])
if J1:
J *= ideal_1
res[j] = R.ideal(J.gens_reduced())
if R(1) in res[j].gens():
bound = j
break
if not groebner or not ring.is_field():
return res
if R.ngens() == 1:
res0 = [gcd(S(p) for p in J.gens()) for J in res]
res1 = []
for p in res0:
if p == 0:
res1.append([R(0)])
res = {j: gcd(S(p) for p in res[j].gens()) for j in range(bound + 1)}
char_var = dict()
strict = True
j = 0
while strict and j <= bound:
if res[j] == 0:
char_var[j] = [R(0)]
else:
fct = [q[0] for q in R(p).factor()]
fct = [q[0] for q in R(res[j]).factor()]
if fct:
res1.append(fct)
return res1
res1 = []
for J in res:
LJ = J.minimal_associated_primes()
char_var[j] = fct
else:
char_var[j] = []
strict = False
j += 1
return char_var
char_var = dict()
strict = True
j = 0
while strict and j <= bound:
LJ = res[j].minimal_associated_primes()
fct = [id.groebner_basis() for id in LJ]
if fct != [(S.one(),)]:
res1.append(fct)
return res1
char_var[j] = fct
if not fct:
strict = False
j += 1
return char_var

def rewriting_system(self):
"""
Expand Down
4 changes: 4 additions & 0 deletions src/sage/matrix/matrix_laurent_mpolynomial_dense.pxd
Original file line number Diff line number Diff line change
@@ -0,0 +1,4 @@
from sage.matrix.matrix_generic_dense cimport Matrix_generic_dense

cdef class Matrix_Laurent_mpolynomial_dense(Matrix_generic_dense):
pass
114 changes: 114 additions & 0 deletions src/sage/matrix/matrix_laurent_mpolynomial_dense.pyx
Original file line number Diff line number Diff line change
@@ -0,0 +1,114 @@
"""
Dense matrices over multivariate polynomials over fields.

AUTHOR:

- Enrique Artal (2023-??): initial version
"""

# *****************************************************************************
# Copyright (C) 2023 Enrique Artal <artal@unizar.es>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# https://www.gnu.org/licenses/
# *****************************************************************************
from sage.matrix.constructor import identity_matrix
from sage.rings.polynomial.laurent_polynomial_ring_base import LaurentPolynomialRing_generic

cdef class Matrix_laurent_mpolynomial_dense(Matrix_generic_dense):
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"""
Dense matrix over a Laurent multivariate polynomial ring over a field.
"""
def laurent_matrix_reduction(self):
"""
From a matrix `self` of Laurent polynomials, apply elementary operations
to obtain a matrix `P` of polynomials such that the variables do not divide
no column and no row.

OUTPUT:

Three matrices `L`, `P`, `R` such that ``self` equals `L P R`, where `L` and
`R` are diagonal with monomial entries.

EXAMPLES:

sage: R.<x, y> = LaurentPolynomialRing(QQ)
sage: L = [1/3*x^-1*y - 6*x^-2*y^2 - 1/2*x^-2*y, 1/5*x + 1/2*y + 1/6]
sage: L += [1/2 - 5*x^-1*y - 2*x^-1, -1/3*y^-2 - 4*x^-1*y^-1 + 11*x^-1*y^-2]
sage: A = matrix(R, 2, L)
sage: lf, P, rg = A.laurent_matrix_reduction()
sage: lf
[ x^-2 0]
[ 0 x^-1*y^-2]
sage: P
[ 1/3*x - 6*y - 1/2 1/5*x^3 + 1/2*x^2*y + 1/6*x^2]
[ 1/2*x*y - 5*y^2 - 2*y -1/3*x - 4*y + 11]
sage: rg
[y 0]
[0 1]
"""
R = self.base_ring()
n_rows, n_cols = self.dimensions()
mat_l = identity_matrix(R, n_rows)
mat_r = identity_matrix(R, n_cols)
res = self.__copy__()
for j, rw in enumerate(res.rows()):
for t in R.gens():
n = min(mon.degree(t) for a in rw for cf, mon in a)
res.rescale_row(j, t ** -n)
mat_l.rescale_col(j, t ** n)
for j, cl in enumerate(res.columns()):
for t in R.gens():
n = min(mon.degree(t) for a in cl for cf, mon in a)
res.rescale_col(j, t ** -n)
mat_r.rescale_row(j, t ** n)
res = res.change_ring(R.polynomial_ring())
return mat_l, res, mat_r

def _fitting_ideal(self, i):
r"""
Return the `i`-th Fitting ideal of the matrix. This is the ideal generated
by the `n - i` minors, where `n` is the number of columns.

INPUT:

``i`` -- an integer

OUTPUT:

An ideal on the base ring.

EXAMPLES::

sage: R.<x,y,z> = LaurentPolynomialRing(QQ)
sage: M = matrix(R, [[2*x^-1-z, 0, y-z^-2, 0], [0, z - y^-1, z - x, 0],[z - y, x^-2 - y, 0, z]])
sage: M
[-z + 2*x^-1 0 y - z^-2 0]
[ 0 z - y^-1 -x + z 0]
[ -y + z -y + x^-2 0 z]
sage: M.fitting_ideal(0)
Ideal (0) of Multivariate Laurent Polynomial Ring in x, y, z over Rational Field
sage: M.fitting_ideal(1) == M._fitting_ideal(1)
True
sage: M.fitting_ideal(1).groebner_basis()
(x^4 - 2*x^3*y - x*z^3 - 4*x^2*y + 8*x*y^2 + 4*x*y*z + 2*z^2 - 8*y,
x*y*z^2 - x*z - 2*y*z + 2,
x^2*z - x*z^2 - 2*x + 2*z,
y^2*z + 1/4*x^2 - 1/2*x*y - 1/4*x*z - y + 1/2)
sage: M.fitting_ideal(2).groebner_basis()
(1,)
sage: M.fitting_ideal(3).groebner_basis()
(1,)
sage: M.fitting_ideal(4).groebner_basis()
(1,)
sage: [R.ideal(M.minors(i)) == M._fitting_ideal(4 - i) for i in range(5)]
[True, True, True, True, True]

"""
R = self.base_ring()
S = R.polynomial_ring()
A = self.laurent_matrix_reduction()[1].change_ring(S)
J = A._fitting_ideal(i)
return J.change_ring(R)
7 changes: 7 additions & 0 deletions src/sage/matrix/matrix_space.py
Original file line number Diff line number Diff line change
Expand Up @@ -306,6 +306,13 @@
pass
else:
return matrix_mpolynomial_dense.Matrix_mpolynomial_dense
elif isinstance(R, sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_mpair) and R.base_ring() in _Fields:
try:
from . import matrix_laurent_mpolynomial_dense
except ImportError:
pass

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else:
return matrix_laurent_mpolynomial_dense.Matrix_laurent_mpolynomial_dense

# The fallback
from sage.matrix.matrix_generic_dense import Matrix_generic_dense
Expand Down
13 changes: 13 additions & 0 deletions src/sage/rings/polynomial/ideal.py
Original file line number Diff line number Diff line change
Expand Up @@ -85,3 +85,16 @@ def groebner_basis(self, algorithm=None):
gb = self.gens_reduced()
from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence_generic
return PolynomialSequence_generic([gb], self.ring(), immutable=True)

def change_ring(self, R):
"""
Coerce an ideal into a new ring.

EXAMPLES::

sage: R.<q> = QQ[]
sage: I = R.ideal([q^2+q-1])
sage: I.change_ring(RR['q'])
Principal ideal (q^2 + q - 1.00000000000000) of Univariate Polynomial Ring in q over Real Field with 53 bits of precision
"""
return R.ideal(self.gens())
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