def generate(R, P, G, B):
while R:
h = normal(F[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(sdp_LM(F[g], u), LM))
P0 = set(p for p in P if monomial_div(sdp_LM(F[p], u), LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if k not in B:
B[k] = monomial_lcm(sdp_LM(F[i], u), sdp_LM(F[j], u))
G = set([ normal(F[g], G - set([g]))[0] for g in G ])
return R, P, G, B
def critical_pair(f, g, ring):
"""
Compute the critical pair corresponding to two labeled polynomials.
A critical pair is a tuple (um, f, vm, g), where um and vm are
terms such that um * f - vm * g is the S-polynomial of f and g (so,
wlog assume um * f > vm * g).
For performance sake, a critical pair is represented as a tuple
(Sign(um * f), um, f, Sign(vm * g), vm, g), since um * f creates
a new, relatively expensive object in memory, whereas Sign(um *
f) and um are lightweight and f (in the tuple) is a reference to
an already existing object in memory.
"""
domain = ring.domain
ltf = Polyn(f).LT
ltg = Polyn(g).LT
lt = (monomial_lcm(ltf[0], ltg[0]), domain.one)
um = term_div(lt, ltf, domain)
vm = term_div(lt, ltg, domain)
# The full information is not needed (now), so only the product
# with the leading term is considered:
fr = lbp_mul_term(lbp(Sign(f), Polyn(f).leading_term, Num(f)), um)
gr = lbp_mul_term(lbp(Sign(g), Polyn(g).leading_term, Num(g)), vm)
# return in proper order, such that the S-polynomial is just
# u_first * f_first - u_second * f_second:
if lbp_cmp(fr, gr) == -1:
return (Sign(gr), vm, g, Sign(fr), um, f)
else:
return (Sign(fr), um, f, Sign(gr), vm, g)
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 generate(R, P, G, B):
while R:
h = normal(F[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(sdp_LM(F[g], u), LM))
P0 = set(p for p in P if monomial_div(sdp_LM(F[p], u), LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if k not in B:
B[k] = monomial_lcm(sdp_LM(F[i], u), sdp_LM(F[j], u))
G = set([normal(F[g], G - set([g]))[0] for g in G])
return R, P, G, B
def sdp_lcm(f, g, u, O, K):
"""
Computes LCM of two polynomials in `K[X]`.
The LCM is computed as the unique generater of the intersection
of the two ideals generated by `f` and `g`. The approach is to
compute a Groebner basis with respect to lexicographic ordering
of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and
then filtering out the solution that doesn't contain `t`.
References
==========
1. [Cox97]_
"""
from sympy.polys.groebnertools import sdp_groebner
if not f or not g:
return []
if sdp_term_p(f) and sdp_term_p(g):
monom = monomial_lcm(sdp_LM(f, u), sdp_LM(g, u))
fc, gc = sdp_LC(f, K), sdp_LC(g, K)
if K.has_Field:
coeff = K.one
else:
coeff = K.lcm(fc, gc)
return [(monom, coeff)]
if not K.has_Field:
lcm = K.one
else:
fc, f = sdp_primitive(f, K)
gc, g = sdp_primitive(g, K)
lcm = K.lcm(fc, gc)
f_terms = tuple( ((1,) + m, c) for m, c in f )
g_terms = tuple( ((0,) + m, c) for m, c in g ) \
+ tuple( ((1,) + m, -c) for m, c in g )
F = sdp_sort(f_terms, lex)
G = sdp_sort(g_terms, lex)
basis = sdp_groebner([F, G], u, lex, K)
H = [ h for h in basis if sdp_indep_p(h, 0, u) ]
if K.is_one(lcm):
h = [ (m[1:], c) for m, c in H[0] ]
else:
h = [ (m[1:], c * lcm) for m, c in H[0] ]
return sdp_sort(h, O)
def sdp_lcm(f, g, u, O, K):
"""
Computes LCM of two polynomials in `K[X]`.
The LCM is computed as the unique generater of the intersection
of the two ideals generated by `f` and `g`. The approach is to
compute a Groebner basis with respect to lexicographic ordering
of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and
then filtering out the solution that doesn't contain `t`.
References
==========
1. [Cox97]_
"""
from sympy.polys.groebnertools import sdp_groebner
if not f or not g:
return []
if sdp_term_p(f) and sdp_term_p(g):
monom = monomial_lcm(sdp_LM(f, u), sdp_LM(g, u))
fc, gc = sdp_LC(f, K), sdp_LC(g, K)
if K.has_Field:
coeff = K.one
else:
coeff = K.lcm(fc, gc)
return [(monom, coeff)]
if not K.has_Field:
lcm = K.one
else:
fc, f = sdp_primitive(f, K)
gc, g = sdp_primitive(g, K)
lcm = K.lcm(fc, gc)
f_terms = tuple(((1, ) + m, c) for m, c in f)
g_terms = tuple( ((0,) + m, c) for m, c in g ) \
+ tuple( ((1,) + m, -c) for m, c in g )
F = sdp_sort(f_terms, lex)
G = sdp_sort(g_terms, lex)
basis = sdp_groebner([F, G], u, lex, K)
H = [h for h in basis if sdp_indep_p(h, 0, u)]
if K.is_one(lcm):
h = [(m[1:], c) for m, c in H[0]]
else:
h = [(m[1:], c * lcm) for m, c in H[0]]
return sdp_sort(h, O)
def groebner_lcm(f, g):
"""
Computes LCM of two polynomials using Groebner bases.
The LCM is computed as the unique generater of the intersection
of the two ideals generated by `f` and `g`. The approach is to
compute a Groebner basis with respect to lexicographic ordering
of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and
then filtering out the solution that doesn't contain `t`.
References
==========
1. [Cox97]_
"""
assert f.ring == g.ring
ring = f.ring
domain = ring.domain
if not f or not g:
return ring.zero
if len(f) <= 1 and len(g) <= 1:
monom = monomial_lcm(f.LM, g.LM)
coeff = domain.lcm(f.LC, g.LC)
return ring.term_new(monom, coeff)
fc, f = f.primitive()
gc, g = g.primitive()
lcm = domain.lcm(fc, gc)
f_terms = [ ((1,) + monom, coeff) for monom, coeff in f.terms() ]
g_terms = [ ((0,) + monom, coeff) for monom, coeff in g.terms() ] \
+ [ ((1,) + monom,-coeff) for monom, coeff in g.terms() ]
t = Dummy("t")
t_ring = ring.clone(symbols=(t,) + ring.symbols, order=lex)
F = t_ring.from_terms(f_terms)
G = t_ring.from_terms(g_terms)
basis = groebner([F, G], t_ring)
def is_independent(h, j):
return all(not monom[j] for monom in h.monoms())
H = [ h for h in basis if is_independent(h, 0) ]
h_terms = [ (monom[1:], coeff*lcm) for monom, coeff in H[0].terms() ]
h = ring.from_terms(h_terms)
return h
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
"""
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 sdp_spoly(p1, p2, u, O, K):
"""
Compute LCM(LM(p1), LM(p2))/LM(p1)*p1 - LCM(LM(p1), LM(p2))/LM(p2)*p2
"""
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 sdm_spoly(f, g, O, K, phantom=None):
"""
Compute the generalized s-polynomial of ``f`` and ``g``.
The ground field is assumed to be ``K``, and monomials ordered according to
``O``.
This is invalid if either of ``f`` or ``g`` is zero.
If the leading terms of `f` and `g` involve different basis elements of
`F`, their s-poly is defined to be zero. Otherwise it is a certain linear
combination of `f` and `g` in which the leading terms cancel.
See [SCA, defn 2.3.6] for details.
If ``phantom`` is not ``None``, it should be a pair of module elements on
which to perform the same operation(s) as on ``f`` and ``g``. The in this
case both results are returned.
Examples
========
>>> from sympy.polys.distributedmodules import sdm_spoly
>>> from sympy.polys import QQ, lex
>>> f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
>>> g = [((2, 3, 0), QQ(1))]
>>> h = [((1, 2, 3), QQ(1))]
>>> sdm_spoly(f, h, lex, QQ)
[]
>>> sdm_spoly(f, g, lex, QQ)
[((1, 2, 1), 1/1)]
"""
if not f or not g:
return sdm_zero()
LM1 = sdm_LM(f)
LM2 = sdm_LM(g)
if LM1[0] != LM2[0]:
return sdm_zero()
LM1 = LM1[1:]
LM2 = LM2[1:]
lcm = monomial_lcm(LM1, LM2)
m1 = monomial_div(lcm, LM1)
m2 = monomial_div(lcm, LM2)
c = K.quo(-sdm_LC(f, K), sdm_LC(g, K))
r1 = sdm_add(sdm_mul_term(f, (m1, K.one), O, K),
sdm_mul_term(g, (m2, c), O, K), O, K)
if phantom is None:
return r1
r2 = sdm_add(sdm_mul_term(phantom[0], (m1, K.one), O, K),
sdm_mul_term(phantom[1], (m2, c), O, K), O, K)
return r1, r2
def spoly(p1, p2):
"""
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 = p1.LM
LM2 = p2.LM
LCM12 = monomial_lcm(LM1, LM2)
m1 = monomial_div(LCM12, LM1)
m2 = monomial_div(LCM12, LM2)
s1 = p1.mul_monom(m1)
s2 = p2.mul_monom(m2)
s = s1 - s2
return s
def sdm_monomial_lcm(A, B):
"""
Return the "least common multiple" of ``A`` and ``B``.
IF `A = M e_j` and `B = N e_j`, where `M` and `N` are polynomial monomials,
this returns `\lcm(M, N) e_j`. Note that ``A`` and ``B`` involve distinct
monomials.
Otherwise the result is undefined.
>>> from sympy.polys.distributedmodules import sdm_monomial_lcm
>>> sdm_monomial_lcm((1, 2, 3), (1, 0, 5))
(1, 2, 5)
"""
return (A[0],) + monomial_lcm(A[1:], B[1:])
def S_poly(tp1, tp2):
"""expv1,p1 = tp1 with expv1 = p1.leading_expv(), p1 monic;
similarly for tp2.
Compute LCM(LM(p1),LM(p2))/LM(p1)*p1 - LCM(LM(p1),LM(p2))/LM(p2)*p2
Throw LPolyOverflowError if bits_exp is too small for the result.
"""
expv1, p1 = tp1
expv2, p2 = tp2
lp = p1.lp
lcm12 = monomial_lcm(expv1, expv2)
m1 = monomial_div(lcm12, expv1)
m2 = monomial_div(lcm12, expv2)
# TODO oprimize
res = Poly(lp)
res.iadd_m_mul_q(p1, (m1, 1))
res.iadd_m_mul_q(p2, (m2, -1))
return res
def prune(P, S, h):
"""
Prune the pair-set by applying the chain criterion
[SCA, remark 2.5.11].
"""
remove = set()
for (a, b, c) in permutations(S, 3):
A = sdm_LM(a)
B = sdm_LM(b)
C = sdm_LM(c)
if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c]:
continue
if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
remove.add((tuple(a), tuple(c)))
return [(f, g) for (f, g) in P if (h not in [f, g]) or \
((tuple(f), tuple(g)) not in remove and \
(tuple(g), tuple(f)) not in remove)]
def S_poly(tp1, tp2):
"""expv1,p1 = tp1 with expv1 = p1.leading_expv(), p1 monic;
similarly for tp2.
Compute LCM(LM(p1),LM(p2))/LM(p1)*p1 - LCM(LM(p1),LM(p2))/LM(p2)*p2
Throw LPolyOverflowError if bits_exp is too small for the result.
"""
expv1, p1 = tp1
expv2, p2 = tp2
lp = p1.lp
lcm12 = monomial_lcm(expv1, expv2)
m1 = monomial_div(lcm12, expv1)
m2 = monomial_div(lcm12, expv2)
# TODO oprimize
res = Poly(lp)
res.iadd_m_mul_q(p1, (m1, 1))
res.iadd_m_mul_q(p2, (m2, -1))
return res
def prune(P, S, h):
"""
Prune the pair-set by applying the chain criterion
[SCA, remark 2.5.11].
"""
remove = set()
retain = set()
for (a, b, c) in permutations(S, 3):
A = sdm_LM(a)
B = sdm_LM(b)
C = sdm_LM(c)
if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c] or \
any(tuple(x) in retain for x in [a, b, c]):
continue
if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
remove.add((tuple(a), tuple(c)))
retain.update([tuple(b), tuple(c), tuple(a)])
return [(f, g) for (f, g) in P if (h not in [f, g]) or \
((tuple(f), tuple(g)) not in remove and \
(tuple(g), tuple(f)) not in remove)]
def sdm_spoly(f, g, O, K):
"""
Compute the generalized s-polynomial of ``f`` and ``g``.
The ground field is assumed to be ``K``, and monomials ordered according to
``O``.
This is invalid if either of ``f`` or ``g`` is zero.
If the leading terms of `f` and `g` involve different basis elements of
`F`, their s-poly is defined to be zero. Otherwise it is a certain linear
combination of `f` and `g` in which the leading terms cancel.
See [SCA, defn 2.3.6] for details.
Examples
========
>>> from sympy.polys.distributedmodules import sdm_spoly
>>> from sympy.polys import QQ, lex
>>> f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
>>> g = [((2, 3, 0), QQ(1))]
>>> h = [((1, 2, 3), QQ(1))]
>>> sdm_spoly(f, h, lex, QQ)
[]
>>> sdm_spoly(f, g, lex, QQ)
[((1, 2, 1), 1/1)]
"""
if not f or not g:
return sdm_zero()
LM1 = sdm_LM(f)
LM2 = sdm_LM(g)
if LM1[0] != LM2[0]:
return sdm_zero()
LM1 = LM1[1:]
LM2 = LM2[1:]
lcm = monomial_lcm(LM1, LM2)
return sdm_add(
sdm_mul_term(f, (monomial_div(lcm, LM1), K.one), O, K),
sdm_mul_term(
g, (monomial_div(lcm, LM2), K.quo(-sdm_LC(f, K), sdm_LC(g, K))), O,
K), O, K)
def sdm_spoly(f, g, O, K):
"""
Compute the generalized s-polynomial of ``f`` and ``g``.
The ground field is assumed to be ``K``, and monomials ordered according to
``O``.
This is invalid if either of ``f`` or ``g`` is zero.
If the leading terms of `f` and `g` involve different basis elements of
`F`, their s-poly is defined to be zero. Otherwise it is a certain linear
combination of `f` and `g` in which the leading terms cancel.
See [SCA, defn 2.3.6] for details.
Examples
========
>>> from sympy.polys.distributedmodules import sdm_spoly
>>> from sympy.polys import QQ, lex
>>> f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
>>> g = [((2, 3, 0), QQ(1))]
>>> h = [((1, 2, 3), QQ(1))]
>>> sdm_spoly(f, h, lex, QQ)
[]
>>> sdm_spoly(f, g, lex, QQ)
[((1, 2, 1), 1/1)]
"""
if not f or not g:
return sdm_zero()
LM1 = sdm_LM(f)
LM2 = sdm_LM(g)
if LM1[0] != LM2[0]:
return sdm_zero()
LM1 = LM1[1:]
LM2 = LM2[1:]
lcm = monomial_lcm(LM1, LM2)
return sdm_add(sdm_mul_term(f, (monomial_div(lcm, LM1), K.one), O, K),
sdm_mul_term(g, (monomial_div(lcm, LM2),
K.quo(-sdm_LC(f, K), sdm_LC(g, K))), O, K),
O, K)
def test_monomial_lcm():
assert monomial_lcm((3,4,1), (1,2,0)) == (3,4,1)
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 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 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 sdm_groebner(G, NF, O, K):
"""
Compute a minimal standard basis of ``G`` with respect to order ``O``.
The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.
The ground field is assumed to be ``K``, and monomials ordered according
to ``O``.
Let `N` denote the submodule generated by elements of `G`. A standard
basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for
any subset `X` of `F`, `in(X)` denotes the submodule generated by the
initial forms of elements of `X`. [SCA, defn 2.3.2]
A standard basis is called minimal if no subset of it is a standard basis.
One may show that standard bases are always generating sets.
Minimal standard bases are not unique. This algorithm computes a
deterministic result, depending on the particular order of `G`.
See [SCA, algorithm 2.3.8, and remark 1.6.3].
"""
# First compute a standard basis
S = [f for f in G if f]
P = list(combinations(S, 2))
def prune(P, S, h):
"""
Prune the pair-set by applying the chain criterion
[SCA, remark 2.5.11].
"""
remove = set()
retain = set()
for (a, b, c) in permutations(S, 3):
A = sdm_LM(a)
B = sdm_LM(b)
C = sdm_LM(c)
if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c] or \
any(tuple(x) in retain for x in [a, b, c]):
continue
if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
remove.add((tuple(a), tuple(c)))
retain.update([tuple(b), tuple(c), tuple(a)])
return [(f, g) for (f, g) in P if (h not in [f, g]) or \
((tuple(f), tuple(g)) not in remove and \
(tuple(g), tuple(f)) not in remove)]
while P:
# TODO better data structures!!!
#print len(P), len(S)
# Use the "normal selection strategy"
lcms = [(i, sdm_LM(f)[:1] + monomial_lcm(sdm_LM(f)[1:], sdm_LM(g)[1:])) for \
i, (f, g) in enumerate(P)]
i = min(lcms, key=lambda x: O(x[1]))[0]
f, g = P.pop(i)
h = NF(sdm_spoly(f, g, O, K), S, O, K)
if h:
S.append(h)
P.extend((h, f) for f in S if sdm_LM(h)[0] == sdm_LM(f)[0])
P = prune(P, S, h)
# Now interreduce it. (TODO again, better data structures)
S = set(tuple(f) for f in S)
for a, b in permutations(S, 2):
A = sdm_LM(list(a))
B = sdm_LM(list(b))
if sdm_monomial_divides(A, B) and b in S and a in S:
S.remove(b)
return sorted((list(f) for f in S),
key=lambda f: O(sdm_LM(f)),
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 lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, f[ip].LM)
return monomial_div(LCMhg, m)
def test_monomial_lcm():
assert monomial_lcm((3, 4, 1), (1, 2, 0)) == (3, 4, 1)
def select(P):
# normal selection strategy
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P, key=lambda pair: order(monomial_lcm(f[pair[0]].LM, f[pair[1]].LM)))
return pr
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 select(P):
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P, key=lambda (i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u))))
return pr
def sdm_groebner(G, NF, O, K):
"""
Compute a minimal standard basis of ``G`` with respect to order ``O``.
The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.
The ground field is assumed to be ``K``, and monomials ordered according
to ``O``.
Let `N` denote the submodule generated by elements of `G`. A standard
basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for
any subset `X` of `F`, `in(X)` denotes the submodule generated by the
initial forms of elements of `X`. [SCA, defn 2.3.2]
A standard basis is called minimal if no subset of it is a standard basis.
One may show that standard bases are always generating sets.
Minimal standard bases are not unique. This algorithm computes a
deterministic result, depending on the particular order of `G`.
See [SCA, algorithm 2.3.8, and remark 1.6.3].
"""
# First compute a standard basis
S = [f for f in G if f]
P = list(combinations(S, 2))
def prune(P, S, h):
"""
Prune the pair-set by applying the chain criterion
[SCA, remark 2.5.11].
"""
remove = set()
retain = set()
for (a, b, c) in permutations(S, 3):
A = sdm_LM(a)
B = sdm_LM(b)
C = sdm_LM(c)
if len(set([A[0], B[0], C[0]])) != 1 or not h in [a, b, c] or \
any(tuple(x) in retain for x in [a, b, c]):
continue
if monomial_divides(B[1:], monomial_lcm(A[1:], C[1:])):
remove.add((tuple(a), tuple(c)))
retain.update([tuple(b), tuple(c), tuple(a)])
return [(f, g) for (f, g) in P if (h not in [f, g]) or \
((tuple(f), tuple(g)) not in remove and \
(tuple(g), tuple(f)) not in remove)]
while P:
# TODO better data structures!!!
#print len(P), len(S)
# Use the "normal selection strategy"
lcms = [(i, sdm_LM(f)[:1] + monomial_lcm(sdm_LM(f)[1:], sdm_LM(g)[1:])) for \
i, (f, g) in enumerate(P)]
i = min(lcms, key=lambda x: O(x[1]))[0]
f, g = P.pop(i)
h = NF(sdm_spoly(f, g, O, K), S, O, K)
if h:
S.append(h)
P.extend((h, f) for f in S if sdm_LM(h)[0] == sdm_LM(f)[0])
P = prune(P, S, h)
# Now interreduce it. (TODO again, better data structures)
S = set(tuple(f) for f in S)
for a, b in permutations(S, 2):
A = sdm_LM(list(a))
B = sdm_LM(list(b))
if sdm_monomial_divides(A, B) and b in S and a in S:
S.remove(b)
return sorted((list(f) for f in S), key=lambda f: O(sdm_LM(f)),
reverse=True)
def select(P):
# select the pair with minimum LCM(LM(f), LM(g))
pr = min(P,
key=lambda
(i, j): O(monomial_lcm(sdp_LM(f[i], u), sdp_LM(f[j], u))))
return pr
