def matrix_fglm(F, u, O_from, O_to, K):
"""
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**
J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
Computation of Zero-dimensional Groebner Bases by Change of
Ordering
J.C. Faugere's lecture notes:
http://www-salsa.lip6.fr/~jcf/Papers/2010_MPRI5e.pdf
"""
old_basis = _basis(F, u, O_from, K)
M = _representing_matrices(old_basis, F, u, O_from, K)
# V contains the normalforms (wrt O_from) of S
S = [(0,) * (u + 1)]
V = [[K.one] + [K.zero] * (len(old_basis) - 1)]
G = []
L = [(i, 0) for i in xrange(u + 1)] # (i, j) corresponds to x_i * S[j]
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
t = L.pop()
P = _identity_matrix(len(old_basis), K)
while True:
s = len(S)
v = _matrix_mul(M[t[0]], V[t[1]], K)
_lambda = _matrix_mul(P, v, K)
if all(_lambda[i] == K.zero for i in xrange(s, len(old_basis))):
# there is a linear combination of v by V
lt = [(_incr_k(S[t[1]], t[0]), K.one)]
rest = sdp_strip(sdp_sort([(S[i], _lambda[i]) for i in xrange(s)], O_to))
g = sdp_sub(lt, rest, u, O_to, K)
if g != []:
G.append(g)
else:
# v is linearly independant from V
P = _update(s, _lambda, P, K)
S.append(_incr_k(S[t[1]], t[0]))
V.append(v)
L.extend([(i, s) for i in xrange(u + 1)])
L = list(set(L))
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
L = [(k, l) for (k, l) in L if all(monomial_div(_incr_k(S[l], k), sdp_LM(g, u)) is None for g in G)]
if not L:
G = [sdp_monic(g, K) for g in G]
return sorted(G, key=lambda g: O_to(sdp_LM(g, u)), reverse=True)
t = L.pop()
def red_groebner(G, u, O, K):
"""
Compute reduced Groebner basis, from BeckerWeispfenning93, p. 216
Selects a subset of generators, that already generate the ideal
and computes a reduced Groebner basis for them.
"""
def reduction(P, u, O, K):
"""
The actual reduction algorithm.
"""
Q = []
for i, p in enumerate(P):
h = sdp_rem(p, P[:i] + P[i + 1 :], u, O, K)
if h != []:
Q.append(h)
return [sdp_monic(p, K) for p in Q]
F = G
H = []
while F:
f0 = F.pop()
if not any(monomial_divides(sdp_LM(f0, u), sdp_LM(f, u)) for f in F + H):
H.append(f0)
# Becker, Weispfenning, p. 217: H is Groebner basis of the ideal generated by G.
return reduction(H, u, O, K)
def is_zero_dimensional(F, u, O, K):
"""
Checks if the ideal generated by ``F`` is zero-dimensional.
The algorithm checks if the set of monomials not divisible by a
leading monomial of any element of ``F`` is bounded. In general
``F`` has to be a Groebner basis w.r.t. ``O`` but if ``True``
is returned, then the ideal is zero-dimensional.
**References**
David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties
and Algorithms, 3rd edition, p. 234
"""
def single_var(m):
n = 0
for e in m:
if e != 0:
n += 1
return n == 1
exponents = (0,) * (u + 1)
for f in F:
if single_var(sdp_LM(f, u)):
exponents = monomial_mul(exponents, sdp_LM(f, O)) # == sum of exponent vectors
product = 1
for e in exponents:
product *= e
# If product == 0, then there's a variable for which there's
# no degree bound.
return product != 0
def f5_reduce(f, B, u, O, K):
"""
F5-reduce a labeled polynomial f by B.
Continously searches for non-zero labeled polynomial h in B, such
that the leading term lt_h of h divides the leading term lt_f of
f and Sign(lt_h * h) < Sign(f). If such a labeled polynomial h is
found, f gets replaced by f - lt_f / lt_h * h. If no such h can be
found or f is 0, f is no further F5-reducible and f gets returned.
A polynomial that is reducible in the usual sense (sdp_rem)
need not be F5-reducible, e.g.:
>>> from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn
>>> from sympy.polys.distributedpolys import sdp_rem
>>> from sympy.polys.monomialtools import lex
>>> from sympy import QQ
>>> f = lbp(sig((1, 1, 1), 4), [((1, 0, 0), QQ(1))], 3)
>>> g = lbp(sig((0, 0, 0), 2), [((1, 0, 0), QQ(1))], 2)
>>> sdp_rem(Polyn(f), [Polyn(g)], 2, lex, QQ)
[]
>>> f5_reduce(f, [g], 2, lex, QQ)
(((1, 1, 1), 4), [((1, 0, 0), 1/1)], 3)
"""
if Polyn(f) == []:
return f
if K.has_Field:
term_div = _term_ff_div
else:
term_div = _term_rr_div
while True:
g = f
for h in B:
if Polyn(h) != []:
if monomial_divides(sdp_LM(Polyn(f), u), sdp_LM(Polyn(h), u)):
t = term_div(
sdp_LT(Polyn(f), u, K), sdp_LT(Polyn(h), u, K), K)
if sig_cmp(sig_mult(Sign(h), t[0]), Sign(f), O) < 0:
# The following check need not be done and is in general slower than without.
#if not is_rewritable_or_comparable(Sign(gp), Num(gp), B, u, K):
hp = lbp_mul_term(h, t, u, O, K)
f = lbp_sub(f, hp, u, O, K)
break
if g == f or Polyn(f) == []:
return f
def sdp_spoly(p1, p2, u, O, K):
"""
Compute LCM(LM(p1), LM(p2))/LM(p1)*p1 - LCM(LM(p1), LM(p2))/LM(p2)*p2
This is the S-poly provided p1 and p2 are monic
"""
LM1 = sdp_LM(p1, u)
LM2 = sdp_LM(p2, u)
LCM12 = monomial_lcm(LM1, LM2)
m1 = monomial_div(LCM12, LM1)
m2 = monomial_div(LCM12, LM2)
s1 = sdp_mul_term(p1, (m1, K.one), u, O, K)
s2 = sdp_mul_term(p2, (m2, K.one), u, O, K)
s = sdp_sub(s1, s2, u, O, K)
return s
def is_minimal(G, u, O, K):
"""
Checks if G is a minimal Groebner basis.
"""
G.sort(key=lambda g: O(sdp_LM(g, u)))
for i, g in enumerate(G):
if sdp_LC(g, K) != K.one:
return False
for h in G[:i] + G[i + 1 :]:
if monomial_divides(sdp_LM(g, u), sdp_LM(h, u)):
return False
return True
def _basis(G, u, O, K):
"""
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)`.
"""
leading_monomials = [sdp_LM(g, u) for g in G]
candidates = [(0,) * (u + 1)]
basis = []
while candidates:
t = candidates.pop()
basis.append(t)
new_candidates = [
_incr_k(t, k)
for k in xrange(u + 1)
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: O(m), reverse=True)
basis = list(set(basis))
return sorted(basis, key=lambda m: O(m))
def is_rewritable_or_comparable(sign, num, B, u, K):
"""
Check if a labeled polynomial is redundant by checking if its
signature and number imply rewritability or comparability.
(sign, num) is comparable if there exists a labeled polynomial
h in B, such that sign[1] (the index) is less than Sign(h)[1]
and sign[0] is divisible by the leading monomial of h.
(sign, num) is rewritable if there exists a labeled polynomial
h in B, such thatsign[1] is equal to Sign(h)[1], num < Num(h)
and sign[0] is divisible by Sign(h)[0].
"""
for h in B:
# comparable
if sign[1] < Sign(h)[1]:
if monomial_divides(sign[0], sdp_LM(Polyn(h), u)):
return True
# rewritable
if sign[1] == Sign(h)[1]:
if num < Num(h):
if monomial_divides(sign[0], Sign(h)[0]):
return True
return False
def normal(g, J):
h = sdp_rem(g, [f[j] for j in J], u, O, K)
if not h:
return None
else:
h = sdp_monic(h, K)
h = tuple(h)
if not h in I:
I[h] = len(f)
f.append(h)
return sdp_LM(h, u), I[h]
def test_sdp_LM():
assert sdp_LM([], 1) == (0, 0)
assert sdp_LM([((1,0), QQ(1,2))], 1) == (1, 0)
assert sdp_LM([((1,1), QQ(1,4)), ((1,0), QQ(1,2))], 1) == (1, 1)
def f5b(F, u, O, K, gens="", verbose=False):
"""
Computes a reduced Groebner basis for the ideal generated by F.
f5b is an implementation of the F5B algorithm by Yao Sun and
Dingkang Wang. Similarly to Buchberger's algorithm, the algorithm
proceeds by computing critical pairs, computing the S-polynomial,
reducing it and adjoining the reduced S-polynomial if it is not 0.
Unlike Buchberger's algorithm, each polynomial contains additional
information, namely a signature and a number. The signature
specifies the path of computation (i.e. from which polynomial in
the original basis was it derived and how), the number says when
the polynomial was added to the basis. With this information it
is (often) possible to decide if an S-polynomial will reduce to
0 and can be discarded.
Optimizations include: Reducing the generators before computing
a Groebner basis, removing redundant critical pairs when a new
polynomial enters the basis and sorting the critical pairs and
the current basis.
Once a Groebner basis has been found, it gets reduced.
** References **
Yao Sun, Dingkang Wang: "A New Proof for the Correctness of F5
(F5-Like) Algorithm", http://arxiv.org/abs/1004.0084 (specifically
v4)
Thomas Becker, Volker Weispfenning, Groebner bases: A computational
approach to commutative algebra, 1993, p. 203, 216
"""
if not K.has_Field:
raise DomainError("can't compute a Groebner basis over %s" % K)
# reduce polynomials (like in Mario Pernici's implementation) (Becker, Weispfenning, p. 203)
B = F
while True:
F = B
B = []
for i in xrange(len(F)):
p = F[i]
r = sdp_rem(p, F[:i], u, O, K)
if r != []:
B.append(r)
if F == B:
break
# basis
B = [lbp(sig((0,) * (u + 1), i + 1), F[i], i + 1) for i in xrange(len(F))]
B.sort(key=lambda f: O(sdp_LM(Polyn(f), u)), reverse=True)
# critical pairs
CP = [critical_pair(B[i], B[j], u, O, K) for i in xrange(len(B)) for j in xrange(i + 1, len(B))]
CP.sort(key=lambda cp: cp_key(cp, O), reverse=True)
k = len(B)
reductions_to_zero = 0
while len(CP):
cp = CP.pop()
# discard redundant critical pairs:
if is_rewritable_or_comparable(cp[0], Num(cp[2]), B, u, K):
continue
if is_rewritable_or_comparable(cp[3], Num(cp[5]), B, u, K):
continue
s = s_poly(cp, u, O, K)
p = f5_reduce(s, B, u, O, K)
p = lbp(Sign(p), sdp_monic(Polyn(p), K), k + 1)
if Polyn(p) != []:
# remove old critical pairs, that become redundant when adding p:
indices = []
for i, cp in enumerate(CP):
if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p], u, K):
indices.append(i)
elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p], u, K):
indices.append(i)
for i in reversed(indices):
del CP[i]
# only add new critical pairs that are not made redundant by p:
for g in B:
if Polyn(g) != []:
cp = critical_pair(p, g, u, O, K)
if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p], u, K):
continue
elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p], u, K):
continue
CP.append(cp)
# sort (other sorting methods/selection strategies were not as successful)
CP.sort(key=lambda cp: cp_key(cp, O), reverse=True)
# insert p into B:
m = sdp_LM(Polyn(p), u)
if O(m) <= O(sdp_LM(Polyn(B[-1]), u)):
B.append(p)
else:
for i, q in enumerate(B):
if O(m) > O(sdp_LM(Polyn(q), u)):
B.insert(i, p)
break
k += 1
# print(len(B), len(CP), "%d critical pairs removed" % len(indices))
else:
reductions_to_zero += 1
if verbose:
print ("%d reductions to zero" % reductions_to_zero)
# reduce Groebner basis:
H = [sdp_monic(Polyn(g), K) for g in B]
H = red_groebner(H, u, O, K)
return sorted(H, key=lambda f: O(sdp_LM(f, u)), reverse=True)
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (not any(
lcm_divides(ipx)
for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or \
monomial_lcm(mg1, mh) == LCM12 or \
monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
def buchberger(f, u, O, K, gens="", verbose=False):
"""
Computes Groebner basis for a set of polynomials in `K[X]`.
Given a set of multivariate polynomials `F`, finds another
set `G`, such that Ideal `F = Ideal G` and `G` is a reduced
Groebner basis.
The resulting basis is unique and has monic generators if the
ground domains is a field. Otherwise the result is non-unique
but Groebner bases over e.g. integers can be computed (if the
input polynomials are monic).
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
**References**
1. [Bose03]_
2. [Giovini91]_
3. [Ajwa95]_
4. [Cox97]_
Algorithm used: an improved version of Buchberger's algorithm
as presented in T. Becker, V. Weispfenning, Groebner Bases: A
Computational Approach to Commutative Algebra, Springer, 1993,
page 232.
Added optional ``gens`` argument to apply :func:`sdp_str` for
the purpose of debugging the algorithm.
"""
if not K.has_Field:
raise DomainError("can't compute a Groebner basis over %s" % K)
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P, key=lambda pair: O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u))))
return pr
def normal(g, J):
h = sdp_rem(g, [f[j] for j in J], u, O, K)
if not h:
return None
else:
h = sdp_monic(h, K)
h = tuple(h)
if not h in I:
I[h] = len(f)
f.append(h)
return sdp_LM(h, u), I[h]
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (
not any(lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)
):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or monomial_lcm(mg1, mh) == LCM12 or monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
# end of update ################################
if not f:
return []
# replace f with a reduced list of initial polynomials; see [BW] page 203
f1 = f[:]
while True:
f = f1[:]
f1 = []
for i in range(len(f)):
p = f[i]
r = sdp_rem(p, f[:i], u, O, K)
if r:
f1.append(sdp_monic(r, K))
if f == f1:
break
f = [tuple(p) for p in f]
I = {} # ip = I[p]; p = f[ip]
F = set() # set of indices of polynomials
G = set() # set of indices of intermediate would-be Groebner basis
CP = set() # set of pairs of indices of critical pairs
for i, h in enumerate(f):
I[h] = i
F.add(i)
#####################################
# algorithm GROEBNERNEWS2 in [BW] page 232
while F:
# select p with minimum monomial according to the monomial ordering O
h = min([f[x] for x in F], key=lambda f: O(sdp_LM(f, u)))
ih = I[h]
F.remove(ih)
G, CP = update(G, CP, ih)
# count the number of critical pairs which reduce to zero
reductions_to_zero = 0
while CP:
ig1, ig2 = select(CP)
CP.remove((ig1, ig2))
h = sdp_spoly(f[ig1], f[ig2], u, O, K)
# ordering divisors is on average more efficient [Cox] page 111
G1 = sorted(G, key=lambda g: O(sdp_LM(f[g], u)))
ht = normal(h, G1)
if ht:
G, CP = update(G, CP, ht[1])
else:
reductions_to_zero += 1
######################################
# now G is a Groebner basis; reduce it
Gr = set()
for ig in G:
ht = normal(f[ig], G - set([ig]))
if ht:
Gr.add(ht[1])
Gr = [list(f[ig]) for ig in Gr]
# order according to the monomial ordering
Gr = sorted(Gr, key=lambda f: O(sdp_LM(f, u)), reverse=True)
if verbose:
print "reductions_to_zero = %d" % reductions_to_zero
return Gr
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P, key=lambda pair: O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u))))
return pr
def buchberger(f, u, O, K, gens='', verbose=False):
"""
Computes Groebner basis for a set of polynomials in `K[X]`.
Given a set of multivariate polynomials `F`, finds another
set `G`, such that Ideal `F = Ideal G` and `G` is a reduced
Groebner basis.
The resulting basis is unique and has monic generators if the
ground domains is a field. Otherwise the result is non-unique
but Groebner bases over e.g. integers can be computed (if the
input polynomials are monic).
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
References
==========
1. [Bose03]_
2. [Giovini91]_
3. [Ajwa95]_
4. [Cox97]_
Algorithm used: an improved version of Buchberger's algorithm
as presented in T. Becker, V. Weispfenning, Groebner Bases: A
Computational Approach to Commutative Algebra, Springer, 1993,
page 232.
Added optional ``gens`` argument to apply :func:`sdp_str` for
the purpose of debugging the algorithm.
"""
if not K.has_Field:
raise DomainError("can't compute a Groebner basis over %s" % K)
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P,
key=lambda pair:
O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u))))
return pr
def normal(g, J):
h = sdp_rem(g, [f[j] for j in J], u, O, K)
if not h:
return None
else:
h = sdp_monic(h, K)
h = tuple(h)
if not h in I:
I[h] = len(f)
f.append(h)
return sdp_LM(h, u), I[h]
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (not any(
lcm_divides(ipx)
for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or \
monomial_lcm(mg1, mh) == LCM12 or \
monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
# end of update ################################
if not f:
return []
# replace f with a reduced list of initial polynomials; see [BW] page 203
f1 = f[:]
while True:
f = f1[:]
f1 = []
for i in range(len(f)):
p = f[i]
r = sdp_rem(p, f[:i], u, O, K)
if r:
f1.append(sdp_monic(r, K))
if f == f1:
break
f = [tuple(p) for p in f]
I = {} # ip = I[p]; p = f[ip]
F = set() # set of indices of polynomials
G = set() # set of indices of intermediate would-be Groebner basis
CP = set() # set of pairs of indices of critical pairs
for i, h in enumerate(f):
I[h] = i
F.add(i)
#####################################
# algorithm GROEBNERNEWS2 in [BW] page 232
while F:
# select p with minimum monomial according to the monomial ordering O
h = min([f[x] for x in F], key=lambda f: O(sdp_LM(f, u)))
ih = I[h]
F.remove(ih)
G, CP = update(G, CP, ih)
# count the number of critical pairs which reduce to zero
reductions_to_zero = 0
while CP:
ig1, ig2 = select(CP)
CP.remove((ig1, ig2))
h = sdp_spoly(f[ig1], f[ig2], u, O, K)
# ordering divisors is on average more efficient [Cox] page 111
G1 = sorted(G, key=lambda g: O(sdp_LM(f[g], u)))
ht = normal(h, G1)
if ht:
G, CP = update(G, CP, ht[1])
else:
reductions_to_zero += 1
######################################
# now G is a Groebner basis; reduce it
Gr = set()
for ig in G:
ht = normal(f[ig], G - set([ig]))
if ht:
Gr.add(ht[1])
Gr = [list(f[ig]) for ig in Gr]
# order according to the monomial ordering
Gr = sorted(Gr, key=lambda f: O(sdp_LM(f, u)), reverse=True)
if verbose:
print 'reductions_to_zero = %d' % reductions_to_zero
return Gr
def f5b(F, u, O, K, gens='', verbose=False):
"""
Computes a reduced Groebner basis for the ideal generated by F.
f5b is an implementation of the F5B algorithm by Yao Sun and
Dingkang Wang. Similarly to Buchberger's algorithm, the algorithm
proceeds by computing critical pairs, computing the S-polynomial,
reducing it and adjoining the reduced S-polynomial if it is not 0.
Unlike Buchberger's algorithm, each polynomial contains additional
information, namely a signature and a number. The signature
specifies the path of computation (i.e. from which polynomial in
the original basis was it derived and how), the number says when
the polynomial was added to the basis. With this information it
is (often) possible to decide if an S-polynomial will reduce to
0 and can be discarded.
Optimizations include: Reducing the generators before computing
a Groebner basis, removing redundant critical pairs when a new
polynomial enters the basis and sorting the critical pairs and
the current basis.
Once a Groebner basis has been found, it gets reduced.
** References **
Yao Sun, Dingkang Wang: "A New Proof for the Correctness of F5
(F5-Like) Algorithm", http://arxiv.org/abs/1004.0084 (specifically
v4)
Thomas Becker, Volker Weispfenning, Groebner bases: A computational
approach to commutative algebra, 1993, p. 203, 216
"""
if not K.has_Field:
raise DomainError("can't compute a Groebner basis over %s" % K)
# reduce polynomials (like in Mario Pernici's implementation) (Becker, Weispfenning, p. 203)
B = F
while True:
F = B
B = []
for i in xrange(len(F)):
p = F[i]
r = sdp_rem(p, F[:i], u, O, K)
if r != []:
B.append(r)
if F == B:
break
# basis
B = [lbp(sig((0, ) * (u + 1), i + 1), F[i], i + 1) for i in xrange(len(F))]
B.sort(key=lambda f: O(sdp_LM(Polyn(f), u)), reverse=True)
# critical pairs
CP = [
critical_pair(B[i], B[j], u, O, K) for i in xrange(len(B))
for j in xrange(i + 1, len(B))
]
CP.sort(key=lambda cp: cp_key(cp, O), reverse=True)
k = len(B)
reductions_to_zero = 0
while len(CP):
cp = CP.pop()
# discard redundant critical pairs:
if is_rewritable_or_comparable(cp[0], Num(cp[2]), B, u, K):
continue
if is_rewritable_or_comparable(cp[3], Num(cp[5]), B, u, K):
continue
s = s_poly(cp, u, O, K)
p = f5_reduce(s, B, u, O, K)
p = lbp(Sign(p), sdp_monic(Polyn(p), K), k + 1)
if Polyn(p) != []:
# remove old critical pairs, that become redundant when adding p:
indices = []
for i, cp in enumerate(CP):
if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p], u, K):
indices.append(i)
elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p], u, K):
indices.append(i)
for i in reversed(indices):
del CP[i]
# only add new critical pairs that are not made redundant by p:
for g in B:
if Polyn(g) != []:
cp = critical_pair(p, g, u, O, K)
if is_rewritable_or_comparable(cp[0], Num(cp[2]), [p], u,
K):
continue
elif is_rewritable_or_comparable(cp[3], Num(cp[5]), [p], u,
K):
continue
CP.append(cp)
# sort (other sorting methods/selection strategies were not as successful)
CP.sort(key=lambda cp: cp_key(cp, O), reverse=True)
# insert p into B:
m = sdp_LM(Polyn(p), u)
if O(m) <= O(sdp_LM(Polyn(B[-1]), u)):
B.append(p)
else:
for i, q in enumerate(B):
if O(m) > O(sdp_LM(Polyn(q), u)):
B.insert(i, p)
break
k += 1
#print(len(B), len(CP), "%d critical pairs removed" % len(indices))
else:
reductions_to_zero += 1
if verbose:
print("%d reductions to zero" % reductions_to_zero)
# reduce Groebner basis:
H = [sdp_monic(Polyn(g), K) for g in B]
H = red_groebner(H, u, O, K)
return sorted(H, key=lambda f: O(sdp_LM(f, u)), reverse=True)
def matrix_fglm(F, u, O_from, O_to, K):
"""
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
==========
J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
Computation of Zero-dimensional Groebner Bases by Change of
Ordering
J.C. Faugere's lecture notes:
http://www-salsa.lip6.fr/~jcf/Papers/2010_MPRI5e.pdf
"""
old_basis = _basis(F, u, O_from, K)
M = _representing_matrices(old_basis, F, u, O_from, K)
# V contains the normalforms (wrt O_from) of S
S = [(0, ) * (u + 1)]
V = [[K.one] + [K.zero] * (len(old_basis) - 1)]
G = []
L = [(i, 0) for i in xrange(u + 1)] # (i, j) corresponds to x_i * S[j]
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
t = L.pop()
P = _identity_matrix(len(old_basis), K)
while True:
s = len(S)
v = _matrix_mul(M[t[0]], V[t[1]], K)
_lambda = _matrix_mul(P, v, K)
if all(_lambda[i] == K.zero for i in xrange(s, len(old_basis))):
# there is a linear combination of v by V
lt = [(_incr_k(S[t[1]], t[0]), K.one)]
rest = sdp_strip(
sdp_sort([(S[i], _lambda[i]) for i in xrange(s)], O_to))
g = sdp_sub(lt, rest, u, O_to, K)
if g != []:
G.append(g)
else:
# v is linearly independant from V
P = _update(s, _lambda, P, K)
S.append(_incr_k(S[t[1]], t[0]))
V.append(v)
L.extend([(i, s) for i in xrange(u + 1)])
L = list(set(L))
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
L = [(k, l) for (k, l) in L if \
all(monomial_div(_incr_k(S[l], k), sdp_LM(g, u)) is None for g in G)]
if not L:
G = [sdp_monic(g, K) for g in G]
return sorted(G, key=lambda g: O_to(sdp_LM(g, u)), reverse=True)
t = L.pop()
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (
not any(lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)
):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or monomial_lcm(mg1, mh) == LCM12 or monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
def test_sdp_LM():
assert sdp_LM([], 1) == (0, 0)
assert sdp_LM([((1, 0), QQ(1, 2))], 1) == (1, 0)
assert sdp_LM([((1, 1), QQ(1, 4)), ((1, 0), QQ(1, 2))], 1) == (1, 1)
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P, key=lambda pair: O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u))))
return pr
