def d_basis(F, strat=True):
r"""
Return the `d`-basis for the Ideal ``F`` as defined in [BW93]_.
INPUT:
- ``F`` - an ideal
- ``strat`` - use update strategy (default: ``True``)
EXAMPLE::
sage: from sage.rings.polynomial.toy_d_basis import d_basis
sage: A.<x,y> = PolynomialRing(ZZ, 2)
sage: f = -y^2 - y + x^3 + 7*x + 1
sage: fx = f.derivative(x)
sage: fy = f.derivative(y)
sage: I = A.ideal([f,fx,fy])
sage: gb = d_basis(I); gb
[x - 2020, y - 11313, 22627]
"""
R = F.ring()
K = R.base_ring()
G = set(inter_reduction(F.gens()))
B = set(filter(lambda (x, y): x != y, [(f1, f2) for f1 in G for f2 in G]))
D = set()
C = set(B)
LCM = R.monomial_lcm
divides = R.monomial_divides
divides_ZZ = lambda x, y: ZZ(x).divides(ZZ(y))
while B != set():
while C != set():
f1, f2 = select(C)
C.remove((f1, f2))
lcm_lmf1_lmf2 = LCM(LM(f1), LM(f2))
if not any( divides(LM(g), lcm_lmf1_lmf2) and \
divides_ZZ( LC(g), LC(f1) ) and \
divides_ZZ( LC(g), LC(f2) ) \
for g in G):
h = gpol(f1, f2)
h0 = h.reduce(G)
if h0.lc() < 0:
h0 *= -1
if not strat:
D = D.union([(g, h0) for g in G])
G.add(h0)
else:
G, D = update(G, D, h0)
G = inter_reduction(G)
f1, f2 = select(B)
B.remove((f1, f2))
h = spol(f1, f2)
h0 = h.reduce(G)
if h0 != 0:
if h0.lc() < 0:
h0 *= -1
if not strat:
D = D.union([(g, h0) for g in G])
G.add(h0)
else:
G, D = update(G, D, h0)
B = B.union(D)
C = D
D = set()
return Sequence(sorted(inter_reduction(G), reverse=True))
def d_basis(F, strat=True):
r"""
Return the `d`-basis for the Ideal ``F`` as defined in [BW93]_.
INPUT:
- ``F`` - an ideal
- ``strat`` - use update strategy (default: ``True``)
EXAMPLE::
sage: from sage.rings.polynomial.toy_d_basis import d_basis
sage: A.<x,y> = PolynomialRing(ZZ, 2)
sage: f = -y^2 - y + x^3 + 7*x + 1
sage: fx = f.derivative(x)
sage: fy = f.derivative(y)
sage: I = A.ideal([f,fx,fy])
sage: gb = d_basis(I); gb
[x - 2020, y - 11313, 22627]
"""
R = F.ring()
K = R.base_ring()
G = set(inter_reduction(F.gens()))
B = set((f1, f2) for f1 in G for f2 in G if f1 != f2)
D = set()
C = set(B)
LCM = R.monomial_lcm
divides = R.monomial_divides
divides_ZZ = lambda x, y: ZZ(x).divides(ZZ(y))
while B!=set():
while C!=set():
f1,f2 = select(C)
C.remove( (f1,f2) )
lcm_lmf1_lmf2 = LCM(LM(f1),LM(f2) )
if not any( divides(LM(g), lcm_lmf1_lmf2) and \
divides_ZZ( LC(g), LC(f1) ) and \
divides_ZZ( LC(g), LC(f2) ) \
for g in G):
h = gpol(f1,f2)
h0 = h.reduce(G)
if h0.lc() < 0:
h0 *= -1
if not strat:
D = D.union( [(g,h0) for g in G] )
G.add(h0)
else:
G, D = update(G,D,h0)
G = inter_reduction(G)
f1,f2 = select(B)
B.remove((f1,f2))
h = spol(f1,f2)
h0 = h.reduce( G )
if h0 != 0:
if h0.lc() < 0:
h0 *= -1
if not strat:
D = D.union( [(g,h0) for g in G] )
G.add( h0 )
else:
G, D = update(G,D,h0)
B = B.union(D)
C = D
D = set()
return Sequence(sorted(inter_reduction(G),reverse=True))