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r""" `J`-ideals of matrices
Let `B` be an `n\times n`-matrix over a principal ideal domain `D`.
For an ideal `J`, the `J`-ideal of `B` is defined to be `N_J(B) = \{ f\in D[X] \mid f(B) \in M_n(J) \}`.
For a prime element `p` of `D` and `t\ge 0`, a `(p^t)`-minimal polynomial of `B` is a monic polynomial `f\in N_{(p^t)}(B)` of minimal degree.
This module computes these minimal polynomials.
Let `p` be a prime element of `D`. Then there is a finite set `\mathcal{S}_p` of positive integers and monic polynomials `\nu_{ps}` for `s\in\mathcal{S}_p` such that for `t\ge 1`,
.. MATH::
N_{(p^t)}(B) = \mu_BD[X] + p^tD[X] + \sum_{\substack{s\in\mathcal{S}_p \\ s \le b(t) }} p^{\max\{0,t-s\}}\nu_{ps}D[X]
holds where `b(t) = \min\{r\in \mathcal{S}_p \mid r \ge s\}`. The degree of `\nu_{ps}` is strictly increasing in `s\in \mathcal{S}_p` and `\nu_{ps}` is a `(p^s)`-minimal polynomial. If `t\le \max\mathcal{S}_p`, then the summand `\mu_BD[X]` can be omitted.
All computations are done by the class :class:`ComputeMinimalPolynomials` where various intermediate results are cached. It provides the following methods:
* :meth:`~ComputeMinimalPolynomials.p_minimal_polynomials` computes `\mathcal{S}_p` and the monic polynomials `\nu_{ps}`.
* :meth:`~ComputeMinimalPolynomials.null_ideal` determines `N_{(p^t)}(B)`.
* :meth:`~ComputeMinimalPolynomials.prime_candidates` determines all primes `p` where `\mathcal{S}_p` might be non-empty.
* :meth:`~ComputeMinimalPolynomials.integer_valued_polynomials_generators` determines the generators of the ring `\{f \in K[X] \mid f(B) \in M_n(D)\}` of integer valued polynomials on `B`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.prime_candidates() [2, 3, 5] sage: for t in range(4): ....: print(C.null_ideal(2^t)) Principal ideal (1) of Univariate Polynomial Ring in x over Integer Ring Ideal (2, x^2 + x) of Univariate Polynomial Ring in x over Integer Ring Ideal (4, x^2 + 3*x + 2) of Univariate Polynomial Ring in x over Integer Ring Ideal (8, x^3 + x^2 - 12*x - 20, 2*x^2 + 6*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: C.p_minimal_polynomials(2) {2: x^2 + 3*x + 2} sage: C.integer_valued_polynomials_generators() (x^3 + x^2 - 12*x - 20, [1, 1/4*x^2 + 3/4*x + 1/2])
The last output means that
.. MATH::
\{f \in \QQ[X] \mid f(B) \in M_3(\ZZ)\} = (x^3 + x^2 - 12x - 20)\QQ[X] + \ZZ[X] + \frac{1}{4}(x^2 + 3x + 2) \ZZ[X].
.. TODO::
Test code over PIDs other than ZZ.
This requires implementation of :meth:`~sage.matrix.matrix_integer_dense.Matrix_integer_dense.frobenius` over more general domains than ZZ.
Additionally, :func:`lifting` requires modification or a bug needs fixing, see `AskSage Question 35555 <https://ask.sagemath.org/question/35555/lifting-a-matrix-from-mathbbqyy-1/>`_.
REFERENCES:
[Ris2016]_, [HR2016]_
AUTHORS:
- Clemens Heuberger (2016) - Roswitha Rissner (2016)
ACKNOWLEDGEMENT:
- Clemens Heuberger is supported by the Austrian Science Fund (FWF): P 24644-N26.
- Roswitha Rissner is supported by the Austrian Science Fund (FWF): P 27816-N26.
Classes and Methods =================== """
# ***************************************************************************** # Copyright (C) 2016 Clemens Heuberger <clemens.heuberger@aau.at> # 2016 Roswitha Rissner <roswitha.rissner@tugraz.at> # # 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/ # *****************************************************************************
r""" Compute generators of `\{f \in D[X]^d \mid Af \equiv 0 \pmod{p^{t}}\}` given generators of `\{f\in D[X]^d \mid Af \equiv 0\pmod{p^{t-1}}\}`.
INPUT:
- ``p`` -- a prime element of some principal ideal domain `D`
- ``t`` -- a non-negative integer
- ``A`` -- a `c\times d` matrix over `D[X]`
- ``G`` -- a matrix over `D[X]`. The columns of `\begin{pmatrix}p^{t-1}I& G\end{pmatrix}` are generators of `\{ f\in D[X]^d \mid Af \equiv 0\pmod{p^{t-1}}\}`; can be set to ``None`` if ``t`` is zero
OUTPUT:
A matrix `F` over `D[X]` such that the columns of `\begin{pmatrix}p^tI&F&pG\end{pmatrix}` are generators of `\{ f\in D[X]^d \mid Af \equiv 0\pmod{p^t}\}`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import lifting sage: X = polygen(ZZ, 'X') sage: A = matrix([[1, X], [2*X, X^2]]) sage: G0 = lifting(5, 0, A, None) sage: G1 = lifting(5, 1, A, G0); G1 [] sage: (A*G1 % 5).is_zero() True sage: A = matrix([[1, X, X^2], [2*X, X^2, 3*X^3]]) sage: G0 = lifting(5, 0, A, None) sage: G1 = lifting(5, 1, A, G0); G1 [3*X^2] [ X] [ 1] sage: (A*G1 % 5).is_zero() True sage: G2 = lifting(5, 2, A, G1); G2 [15*X^2 23*X^2] [ 5*X X] [ 5 1] sage: (A*G2 % 25).is_zero() True sage: lifting(5, 10, A, G1) Traceback (most recent call last): ... ValueError: A*G not zero mod 5^9
ALGORITHM:
[HR2016]_, Algorithm 1.
TESTS::
sage: A = matrix([[1, X], [X, X^2]]) sage: G0 = lifting(5, 0, A, None) sage: G1 = lifting(5, 1, A, G0); G1 Traceback (most recent call last): ... ValueError: [ 1 X|] [ X X^2|] does not have full rank. """
#assert Sb * ARb * Tb == Db #assert all(i == j or Db[i, j].is_zero() # for i in range(Db.nrows()) # for j in range(Db.ncols()))
r""" Compute the `p`-part of a polynomial.
INPUT:
- ``f`` -- a polynomial over `D`
- ``p`` -- a prime in `D`
OUTPUT:
A polynomial `g` such that `\deg g \le \deg f` and all non-zero coefficients of `f - p g` are not divisible by `p`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import p_part sage: X = polygen(ZZ, 'X') sage: f = X^3 + 5*X + 25 sage: g = p_part(f, 5); g X + 5 sage: f - 5*g X^3
TESTS:
Return value is supposed to be a polynomial, see :trac:`22402`
sage: g = p_part(X+1, 2) sage: g.parent() Univariate Polynomial Ring in X over Integer Ring
""" if c % p == 0))
r""" Create an object for computing `(p^t)`-minimal polynomials and `J`-ideals.
For an ideal `J` and a square matrix `B` over a principal ideal domain `D`, the `J`-ideal of `B` is defined to be `N_J(B) = \{ f\in D[X] \mid f(B) \in M_n(J) \}`.
For a prime element `p` of `D` and `t\ge 0`, a `(p^t)`-minimal polynomial of `B` is a monic polynomial `f\in N_{(p^t)}(B)` of minimal degree.
The characteristic polynomial of `B` is denoted by `\chi_B`; `n` is the size of `B`.
INPUT:
- ``B`` -- a square matrix over a principal ideal domain `D`
OUTPUT:
An object which allows to call :meth:`p_minimal_polynomials`, :meth:`null_ideal` and :meth:`integer_valued_polynomials_generators`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.prime_candidates() [2, 3, 5] sage: for t in range(4): ....: print(C.null_ideal(2^t)) Principal ideal (1) of Univariate Polynomial Ring in x over Integer Ring Ideal (2, x^2 + x) of Univariate Polynomial Ring in x over Integer Ring Ideal (4, x^2 + 3*x + 2) of Univariate Polynomial Ring in x over Integer Ring Ideal (8, x^3 + x^2 - 12*x - 20, 2*x^2 + 6*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: C.p_minimal_polynomials(2) {2: x^2 + 3*x + 2} sage: C.integer_valued_polynomials_generators() (x^3 + x^2 - 12*x - 20, [1, 1/4*x^2 + 3/4*x + 1/2]) """ r""" Initialize the ComputeMinimalPolynomials class.
INPUT:
- ``B`` -- a square matrix
TESTS::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: ComputeMinimalPolynomials(matrix([[1, 2]])) Traceback (most recent call last): ... TypeError: square matrix required """
r""" Replace possibly non-monic generators of `N_{(p^t)}(B)` by monic generators.
INPUT:
- ``p`` -- a prime element of `D`
- ``t`` -- a non-negative integer
- ``pt_generators`` -- a list `(g_1, \ldots, g_s)` of polynomials in `D[X]` such that `N_{(p^t)}(B) = (g_1, \ldots, g_s) + pN_{(p^{t-1})}(B)`
- ``prev_nu`` -- a `(p^{t-1})`-minimal polynomial of `B`
OUTPUT:
A list `(h_1, \ldots, h_r)` of monic polynomials such that `N_{(p^t)}(B) = (h_1, \ldots, h_r) + pN_{(p^{t-1})}(B)`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: x = polygen(ZZ, 'x') sage: nu_1 = x^2 + x sage: generators_4 = [2*x^2 + 2*x, x^2 + 3*x + 2] sage: C.find_monic_replacements(2, 2, generators_4, nu_1) [x^2 + 3*x + 2]
TESTS::
sage: C.find_monic_replacements(2, 3, generators_4, nu_1) Traceback (most recent call last): ... ValueError: [2*x^2 + 2*x, x^2 + 3*x + 2] not in N_{(2^3)}(B) sage: C.find_monic_replacements(2, 2, generators_4, x^2) Traceback (most recent call last): ... ValueError: x^2 not in N_{(2^1)}(B)
ALGORITHM:
[HR2016]_, Algorithms 2 and 3. """
for g in pt_generators): (pt_generators, p, t))
#reduce coefficients mod p^t to keep coefficients small
"Something went wrong in find_monic_replacements"
r""" Compute `(p^t)`-minimal polynomial of `B`.
INPUT:
- ``p`` -- a prime element of `D`
- ``t`` -- a positive integer
- ``pt_generators`` -- a list `(g_1, \ldots, g_s)` of polynomials in `D[X]` such that `N_{(p^t)}(B) = (g_1, \ldots, g_s) + pN_{(p^{t-1})}(B)`
- ``prev_nu`` -- a `(p^{t-1})`-minimal polynomial of `B`
OUTPUT:
A `(p^t)`-minimal polynomial of `B`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: x = polygen(ZZ, 'x') sage: nu_1 = x^2 + x sage: generators_4 = [2*x^2 + 2*x, x^2 + 3*x + 2] sage: C.current_nu(2, 2, generators_4, nu_1) x^2 + 3*x + 2
ALGORITHM:
[HR2016]_, Algorithm 4.
TESTS::
sage: C.current_nu(2, 3, generators_4, nu_1) Traceback (most recent call last): ... ValueError: [2*x^2 + 2*x, x^2 + 3*x + 2] not in N_{(2^3)}(B) sage: C.current_nu(2, 2, generators_4, x^2) Traceback (most recent call last): ... ValueError: x^2 not in N_{(2^1)}(B) """
for g in pt_generators): (pt_generators, p, t))
# find poly of minimal degree
# find nu #take first element in generators not equal g heapq.heappush(heap, (h.degree(), h)) deg_g, g = heapq.heappushpop(heap, (deg_g, g))
r""" Compute matrix for McCoy's criterion.
INPUT:
- ``p`` -- a prime element in `D`
- ``t`` -- a positive integer
- ``nu`` -- a `(p^t)`-minimal polynomial of `B`
OUTPUT:
An `(n^2 + 1) \times 1` matrix `g` with first entry ``nu`` such that `\begin{pmatrix}b& -\chi_B I\end{pmatrix}g \equiv 0\pmod{p^t}` where `b` consists of the entries of `\operatorname{adj}(X-B)`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: x = polygen(ZZ, 'x') sage: nu_4 = x^2 + 3*x + 2 sage: g = C.mccoy_column(2, 2, nu_4) sage: b = matrix(9, 1, (x-B).adjoint().list()) sage: M = matrix.block([[b , -B.charpoly(x)*matrix.identity(9)]]) sage: (M*g % 4).is_zero() True
ALGORITHM:
[HR2016]_, Algorithm 5.
TESTS::
sage: nu_2 = x^2 + x sage: C.mccoy_column(2, 2, nu_2) Traceback (most recent call last): ... ValueError: x^2 + x not in (2^2)-ideal
""" "%s not in (%s^%s)-ideal" % (nu, p, t))
[nu] + [(nu*b).quo_rem(self.chi_B)[0] for b in self._A[:, 0].list()])
"McCoy column incorrect"
r""" Compute `(p^s)`-minimal polynomials `\nu_s` of `B`.
Compute a finite subset `\mathcal{S}` of the positive integers and `(p^s)`-minimal polynomials `\nu_s` for `s \in \mathcal{S}`.
For `0 < t \le \max \mathcal{S}`, a `(p^t)`-minimal polynomial is given by `\nu_s` where `s = \min\{ r \in \mathcal{S} \mid r\ge t \}`. For `t > \max \mathcal{S}`, the minimal polynomial of `B` is also a `(p^t)`-minimal polynomial.
INPUT:
- ``p`` -- a prime in `D`
- ``s_max`` -- a positive integer (default: ``None``); if set, only `(p^s)`-minimal polynomials for ``s <= s_max`` are computed (see below for details)
OUTPUT:
A dictionary. Keys are the finite set `\mathcal{S}`, the values are the associated `(p^s)`-minimal polynomials `\nu_s`, `s \in \mathcal{S}`.
Setting ``s_max`` only affects the output if ``s_max`` is at most `\max\mathcal{S}` where `\mathcal{S}` denotes the full set. In that case, only those `\nu_s` with ``s <= s_max`` are returned where ``s_max`` is always included even if it is not included in the full set `\mathcal{S}`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.p_minimal_polynomials(2) {2: x^2 + 3*x + 2} sage: set_verbose(1) sage: C = ComputeMinimalPolynomials(B) sage: C.p_minimal_polynomials(2) verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) p = 2, t = 1: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) Result of lifting: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) F = verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) [x^2 + x] [ x] [ 0] [ 1] [ 1] [ x + 1] [ 1] [ 0] [ 0] [ x + 1] verbose 1 (...: compute_J_ideal.py, current_nu) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, current_nu) (x^2 + x) verbose 1 (...: compute_J_ideal.py, current_nu) Generators with (p^t)-generating property: verbose 1 (...: compute_J_ideal.py, current_nu) [x^2 + x] verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) nu = x^2 + x verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) corresponding columns for G verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) [x^2 + x] [ x + 2] [ 0] [ 1] [ 1] [ x - 1] [ -1] [ 10] [ 0] [ x + 1] verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) p = 2, t = 2: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) Result of lifting: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) F = verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) [ 2*x^2 + 2*x x^2 + 3*x + 2] [ 2*x x + 4] [ 0 0] [ 2 1] [ 2 1] [ 2*x + 2 x + 1] [ 2 -1] [ 0 10] [ 0 0] [ 2*x + 2 x + 3] verbose 1 (...: compute_J_ideal.py, current_nu) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, current_nu) (2*x^2 + 2*x, x^2 + 3*x + 2) verbose 1 (...: compute_J_ideal.py, current_nu) Generators with (p^t)-generating property: verbose 1 (...: compute_J_ideal.py, current_nu) [x^2 + 3*x + 2] verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) nu = x^2 + 3*x + 2 verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) corresponding columns for G verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) [x^2 + 3*x + 2] [ x + 4] [ 0] [ 1] [ 1] [ x + 1] [ -1] [ 10] [ 0] [ x + 3] verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) p = 2, t = 3: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) Result of lifting: verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) F = verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) [x^3 + 7*x^2 + 6*x x^3 + 3*x^2 + 2*x] [ x^2 + 8*x x^2 + 4*x] [ 0 0] [ x x + 4] [ x + 4 x] [ x^2 + 5*x + 4 x^2 + x] [ -x + 4 -x] [ 10*x 10*x] [ 0 0] [ x^2 + 7*x x^2 + 3*x + 4] verbose 1 (...: compute_J_ideal.py, current_nu) ------------------------------------------ verbose 1 (...: compute_J_ideal.py, current_nu) (x^3 + 7*x^2 + 6*x, x^3 + 3*x^2 + 2*x) verbose 1 (...: compute_J_ideal.py, current_nu) Generators with (p^t)-generating property: verbose 1 (...: compute_J_ideal.py, current_nu) [x^3 + 7*x^2 + 6*x, x^3 + 3*x^2 + 2*x] verbose 1 (...: compute_J_ideal.py, current_nu) [x^3 + 3*x^2 + 2*x] verbose 1 (...: compute_J_ideal.py, p_minimal_polynomials) nu = x^3 + 3*x^2 + 2*x {2: x^2 + 3*x + 2} sage: set_verbose(0) sage: C.p_minimal_polynomials(2, s_max=1) {1: x^2 + x} sage: C.p_minimal_polynomials(2, s_max=2) {2: x^2 + 3*x + 2} sage: C.p_minimal_polynomials(2, s_max=3) {2: x^2 + 3*x + 2}
ALGORITHM:
[HR2016]_, Algorithm 5. """
else:
for r, polynomial in iteritems(p_min_polys) if r < s_max}
r""" Return the `(b)`-ideal `N_{(b)}(B)=\{f\in D[X] \mid f(B)\in M_n(bD)\}`.
INPUT:
- ``b`` -- an element of `D` (default: 0)
OUTPUT:
An ideal in `D[X]`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.null_ideal() Principal ideal (x^3 + x^2 - 12*x - 20) of Univariate Polynomial Ring in x over Integer Ring sage: C.null_ideal(2) Ideal (2, x^2 + x) of Univariate Polynomial Ring in x over Integer Ring sage: C.null_ideal(4) Ideal (4, x^2 + 3*x + 2) of Univariate Polynomial Ring in x over Integer Ring sage: C.null_ideal(8) Ideal (8, x^3 + x^2 - 12*x - 20, 2*x^2 + 6*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: C.null_ideal(3) Ideal (3, x^3 + x^2 - 12*x - 20) of Univariate Polynomial Ring in x over Integer Ring sage: C.null_ideal(6) Ideal (6, 2*x^3 + 2*x^2 - 24*x - 40, 3*x^2 + 3*x) of Univariate Polynomial Ring in x over Integer Ring """
else: for s, nu in iteritems(p_polynomials)]
"Polynomials not in %s-ideal" % (b,)
r""" Determine those primes `p` where `\mu_B` might not be a `(p)`-minimal polynomial.
OUTPUT:
A list of primes.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.prime_candidates() [2, 3, 5] sage: C.p_minimal_polynomials(2) {2: x^2 + 3*x + 2} sage: C.p_minimal_polynomials(3) {} sage: C.p_minimal_polynomials(5) {}
This means that `3` and `5` were candidates, but actually, `\mu_B` turns out to be a `(3)`-minimal polynomial and a `(5)`-minimal polynomial. """
r""" Determine the generators of the ring of integer valued polynomials on `B`.
OUTPUT:
A pair ``(mu_B, P)`` where ``P`` is a list of polynomials in `K[X]` such that
.. MATH::
\{f \in K[X] \mid f(B) \in M_n(D)\} = \mu_B K[X] + \sum_{g\in P} g D[X]
where `K` denotes the fraction field of `D`.
EXAMPLES::
sage: from sage.matrix.compute_J_ideal import ComputeMinimalPolynomials sage: B = matrix(ZZ, [[1, 0, 1], [1, -2, -1], [10, 0, 0]]) sage: C = ComputeMinimalPolynomials(B) sage: C.integer_valued_polynomials_generators() (x^3 + x^2 - 12*x - 20, [1, 1/4*x^2 + 3/4*x + 1/2]) """ [nu/p**s for p in self.prime_candidates() for s, nu in iteritems(self.p_minimal_polynomials(p))]) |