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 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]
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]
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 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 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 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 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 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()