def is_rewritable_or_comparable(sign, num, B):
"""
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(Polyn(h).LM, sign[0]):
return True
# rewritable
if sign[1] == Sign(h)[1]:
if num < Num(h):
if monomial_divides(Sign(h)[0], sign[0]):
return True
return False
def is_rewritable_or_comparable(sign, num, B):
"""
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(Polyn(h).LM, sign[0]):
return True
# rewritable
if sign[1] == Sign(h)[1]:
if num < Num(h):
if monomial_divides(Sign(h)[0], sign[0]):
return True
return False
def red_groebner(G, ring):
"""
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):
"""
The actual reduction algorithm.
"""
Q = []
for i, p in enumerate(P):
h = p.rem(P[:i] + P[i + 1:])
if h:
Q.append(h)
return [p.monic() for p in Q]
F = G
H = []
while F:
f0 = F.pop()
if not any(monomial_divides(f.LM, f0.LM) 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)
def red_groebner(G, ring):
"""
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):
"""
The actual reduction algorithm.
"""
Q = []
for i, p in enumerate(P):
h = p.rem(P[:i] + P[i + 1:])
if h:
Q.append(h)
return [p.monic() for p in Q]
F = G
H = []
while F:
f0 = F.pop()
if not any(monomial_divides(f.LM, f0.LM) 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)
def f5_reduce(f, B):
"""
F5-reduce a labeled polynomial f by B.
Continuously 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 need not be
F5-reducible, e.g.:
>>> from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn
>>> from sympy.polys import ring, QQ, lex
>>> R, x,y,z = ring("x,y,z", QQ, lex)
>>> f = lbp(sig((1, 1, 1), 4), x, 3)
>>> g = lbp(sig((0, 0, 0), 2), x, 2)
>>> Polyn(f).rem([Polyn(g)])
0
>>> f5_reduce(f, [g])
(((1, 1, 1), 4), x, 3)
"""
order = Polyn(f).ring.order
domain = Polyn(f).ring.domain
if not Polyn(f):
return f
while True:
g = f
for h in B:
if Polyn(h):
if monomial_divides(Polyn(h).LM, Polyn(f).LM):
t = term_div(Polyn(f).LT, Polyn(h).LT, domain)
if sig_cmp(sig_mult(Sign(h), t[0]), Sign(f), order) < 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):
hp = lbp_mul_term(h, t)
f = lbp_sub(f, hp)
break
if g == f or not Polyn(f):
return f
def f5_reduce(f, B):
"""
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 need not be
F5-reducible, e.g.:
>>> from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn
>>> from sympy.polys import ring, QQ, lex
>>> R, x,y,z = ring("x,y,z", QQ, lex)
>>> f = lbp(sig((1, 1, 1), 4), x, 3)
>>> g = lbp(sig((0, 0, 0), 2), x, 2)
>>> Polyn(f).rem([Polyn(g)])
0
>>> f5_reduce(f, [g])
(((1, 1, 1), 4), x, 3)
"""
order = Polyn(f).ring.order
domain = Polyn(f).ring.domain
if not Polyn(f):
return f
while True:
g = f
for h in B:
if Polyn(h):
if monomial_divides(Polyn(h).LM, Polyn(f).LM):
t = term_div(Polyn(f).LT, Polyn(h).LT, domain)
if sig_cmp(sig_mult(Sign(h), t[0]), Sign(f), order) < 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):
hp = lbp_mul_term(h, t)
f = lbp_sub(f, hp)
break
if g == f or not Polyn(f):
return f
def is_minimal(G, ring):
"""
Checks if G is a minimal Groebner basis.
"""
order = ring.order
domain = ring.domain
G.sort(key=lambda g: order(g.LM))
for i, g in enumerate(G):
if g.LC != domain.one:
return False
for h in G[:i] + G[i + 1:]:
if monomial_divides(h.LM, g.LM):
return False
return True
def is_minimal(G, ring):
"""
Checks if G is a minimal Groebner basis.
"""
order = ring.order
domain = ring.domain
G.sort(key=lambda g: order(g.LM))
for i, g in enumerate(G):
if g.LC != domain.one:
return False
for h in G[:i] + G[i + 1:]:
if monomial_divides(h.LM, g.LM):
return False
return True
def is_reduced(G, ring):
"""
Checks if G is a reduced Groebner basis.
"""
order = ring.order
domain = ring.domain
G.sort(key=lambda g: order(g.LM))
for i, g in enumerate(G):
if g.LC != domain.one:
return False
for term in g.terms():
for h in G[:i] + G[i + 1:]:
if monomial_divides(h.LM, term[0]):
return False
return True
def _compute_smd(sub, mlist: list):
return (monomial_deg(sub) - 1) * len(list(filter(lambda m: monomial_divides(sub, m), mlist)))
def test_monomial_divides():
assert monomial_divides((1, 2, 3), (4, 5, 6)) is True
assert monomial_divides((1, 2, 3), (0, 5, 6)) is False
def test_monomial_divides():
assert monomial_divides((1, 2, 3), (4, 5, 6)) is True
assert monomial_divides((1, 2, 3), (0, 5, 6)) is False
