"""Implementation of matrix FGLM Groebner basis conversion algorithm. """
from __future__ import print_function, division
from sympy.polys.monomials import monomial_mul, monomial_div
from sympy.core.compatibility import range
[docs]def matrix_fglm(F, ring, O_to):
"""
Converts the reduced Groebner basis ``F`` of a zero-dimensional
ideal w.r.t. ``O_from`` to a reduced Groebner basis
w.r.t. ``O_to``.
References
==========
.. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
Computation of Zero-dimensional Groebner Bases by Change of
Ordering
"""
domain = ring.domain
ngens = ring.ngens
ring_to = ring.clone(order=O_to)
old_basis = _basis(F, ring)
M = _representing_matrices(old_basis, F, ring)
# V contains the normalforms (wrt O_from) of S
S = [ring.zero_monom]
V = [[domain.one] + [domain.zero] * (len(old_basis) - 1)]
G = []
L = [(i, 0) for i in range(ngens)] # (i, j) corresponds to x_i * S[j]
L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True)
t = L.pop()
P = _identity_matrix(len(old_basis), domain)
while True:
s = len(S)
v = _matrix_mul(M[t[0]], V[t[1]])
_lambda = _matrix_mul(P, v)
if all(_lambda[i] == domain.zero for i in range(s, len(old_basis))):
# there is a linear combination of v by V
lt = ring.term_new(_incr_k(S[t[1]], t[0]), domain.one)
rest = ring.from_dict({S[i]: _lambda[i] for i in range(s)})
g = (lt - rest).set_ring(ring_to)
if g:
G.append(g)
else:
# v is linearly independent from V
P = _update(s, _lambda, P)
S.append(_incr_k(S[t[1]], t[0]))
V.append(v)
L.extend([(i, s) for i in range(ngens)])
L = list(set(L))
L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True)
L = [(k, l) for (k, l) in L if all(monomial_div(_incr_k(S[l], k), g.LM) is None for g in G)]
if not L:
G = [ g.monic() for g in G ]
return sorted(G, key=lambda g: O_to(g.LM), reverse=True)
t = L.pop()
def _incr_k(m, k):
return tuple(list(m[:k]) + [m[k] + 1] + list(m[k + 1:]))
def _identity_matrix(n, domain):
M = [[domain.zero]*n for _ in range(n)]
for i in range(n):
M[i][i] = domain.one
return M
def _matrix_mul(M, v):
return [sum([row[i] * v[i] for i in range(len(v))]) for row in M]
def _update(s, _lambda, P):
"""
Update ``P`` such that for the updated `P'` `P' v = e_{s}`.
"""
k = min([j for j in range(s, len(_lambda)) if _lambda[j] != 0])
for r in range(len(_lambda)):
if r != k:
P[r] = [P[r][j] - (P[k][j] * _lambda[r]) / _lambda[k] for j in range(len(P[r]))]
P[k] = [P[k][j] / _lambda[k] for j in range(len(P[k]))]
P[k], P[s] = P[s], P[k]
return P
def _representing_matrices(basis, G, ring):
r"""
Compute the matrices corresponding to the linear maps `m \mapsto
x_i m` for all variables `x_i`.
"""
domain = ring.domain
u = ring.ngens-1
def var(i):
return tuple([0] * i + [1] + [0] * (u - i))
def representing_matrix(m):
M = [[domain.zero] * len(basis) for _ in range(len(basis))]
for i, v in enumerate(basis):
r = ring.term_new(monomial_mul(m, v), domain.one).rem(G)
for monom, coeff in r.terms():
j = basis.index(monom)
M[j][i] = coeff
return M
return [representing_matrix(var(i)) for i in range(u + 1)]
def _basis(G, ring):
r"""
Computes a list of monomials which are not divisible by the leading
monomials wrt to ``O`` of ``G``. These monomials are a basis of
`K[X_1, \ldots, X_n]/(G)`.
"""
order = ring.order
leading_monomials = [g.LM for g in G]
candidates = [ring.zero_monom]
basis = []
while candidates:
t = candidates.pop()
basis.append(t)
new_candidates = [_incr_k(t, k) for k in range(ring.ngens)
if all(monomial_div(_incr_k(t, k), lmg) is None
for lmg in leading_monomials)]
candidates.extend(new_candidates)
candidates.sort(key=lambda m: order(m), reverse=True)
basis = list(set(basis))
return sorted(basis, key=lambda m: order(m))
