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 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 critical_pair(f, g, u, O, K):
"""
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.
"""
ltf = sdp_LT(Polyn(f), u, K)
ltg = sdp_LT(Polyn(g), u, K)
lt = (monomial_lcm(ltf[0], ltg[0]), K.one)
if K.has_Field:
term_div = _term_ff_div
else:
term_div = _term_rr_div
um = term_div(lt, ltf, K)
vm = term_div(lt, ltg, K)
# The full information is not needed (now), so only the product
# with the leading term is considered:
fr = lbp_mul_term(lbp(Sign(f), [sdp_LT(Polyn(f), u, K)], Num(f)), um, u, O,
K)
gr = lbp_mul_term(lbp(Sign(g), [sdp_LT(Polyn(g), u, K)], Num(g)), vm, u, O,
K)
# return in proper order, such that the S-polynomial is just
# u_first * f_first - u_second * f_second:
if lbp_cmp(fr, gr, O) == -1:
return (Sign(gr), vm, g, Sign(fr), um, f)
else:
return (Sign(fr), um, f, Sign(gr), vm, g)
def critical_pair(f, g, u, O, K):
"""
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.
"""
ltf = sdp_LT(Polyn(f), u, K)
ltg = sdp_LT(Polyn(g), u, K)
lt = (monomial_lcm(ltf[0], ltg[0]), K.one)
if K.has_Field:
term_div = _term_ff_div
else:
term_div = _term_rr_div
um = term_div(lt, ltf, K)
vm = term_div(lt, ltg, K)
# The full information is not needed (now), so only the product
# with the leading term is considered:
fr = lbp_mul_term(
lbp(Sign(f), [sdp_LT(Polyn(f), u, K)], Num(f)), um, u, O, K)
gr = lbp_mul_term(
lbp(Sign(g), [sdp_LT(Polyn(g), u, K)], Num(g)), vm, u, O, K)
# return in proper order, such that the S-polynomial is just
# u_first * f_first - u_second * f_second:
if lbp_cmp(fr, gr, O) == -1:
return (Sign(gr), vm, g, Sign(fr), um, f)
else:
return (Sign(fr), um, f, Sign(gr), vm, g)
def test_sdp_LT():
assert sdp_LT([], 1, QQ) == ((0, 0), QQ(0))
assert sdp_LT([((1, 0), QQ(1, 2))], 1, QQ) == ((1, 0), QQ(1, 2))
assert sdp_LT(
[((1, 1), QQ(1, 4)), ((1, 0), QQ(1, 2))], 1, QQ) == ((1, 1), QQ(1, 4))
