def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u))))
return pr
def sdp_groebner(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 = minkey(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], 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 = minkey([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 sdp_groebner(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):
# select the pair with minimum LCM(LM(f), LM(g))
pr = minkey(P,
key=lambda
(i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], 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 = 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(r)
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 = minkey([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)
ht = normal(h, G)
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 = [sdp_monic(list(f[ig]), K) 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 select(P):
# select the pair with minimum LCM(LM(f), LM(g))
pr = minkey(P,
key=lambda
(i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u))))
return pr
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = minkey(P, key=lambda pair: O(monomial_lcm(sdp_LM(f[pair[0]], u), sdp_LM(f[pair[1]], u))))
return pr
