# Revision history [back]

It seems like there is no such option. However we can program it ourselves:

from sage.misc.verbose import verbose

def my_reduce(f, G):
verbose('Reducing {} with respect to {}'.format(f, G), level=1)
p = f
r = 0
while not p.is_zero():
lt = p.lt()
candidate = next((g for g in G if g.lt().divides(lt)), None)
if candidate is not None:
coeff = lt // candidate.lt()
verbose('Subtracting {} * ({})'.format(coeff, candidate), level=1)
p -= coeff*candidate
else:
verbose('Term {} cannot be eliminated'.format(p.lt()), level=1)
r += p.lt()
p -= p.lt()
return r

def my_interreduce_groebner_basis(G):
verbose('Interreducing the Groebner basis {}'.format(G), level=1)
i = 0
while i < len(G):
j = 0
while j < len(G):
if i != j and G[i].lt().divides(G[j].lt()):
verbose('Removing {} since its leading term is divisible by that of {}'.format(G[j],G[i]), level=1)
G.pop(j)
else:
j += 1
i += 1
verbose('Normalizing:', level=1)
for i in range(len(G)):
if G[i].lc() != 1:
verbose('Dividing {} by {}'.format(G[i], G[i].lc()), level=1)
G[i] = G[i] / G[i].lc()
verbose('Reducing each element with respect to the rest:', level=1)
for i in range(len(G)):
G[i] = my_reduce(G[i], [G[j] for j in range(len(G)) if j != i])
return G


Example:

sage: R.<x,y> = PolynomialRing(QQ,order='deglex')
sage: G = [y^4 + x^2*y + y^3 - x*y, x*y + y^2, x*y^2 + y^3]
sage: R.ideal(G).basis_is_groebner()
True
sage: set_verbose(1)
sage: my_interreduce_groebner_basis(G)
verbose 1 (my_interreduce_groebner_basis) Interreducing the Groebner basis [y^4 + x^2*y + y^3 - x*y, x*y + y^2, x*y^2 + y^3]
verbose 1 (my_interreduce_groebner_basis) Removing x*y^2 + y^3 since its leading term is divisible by that of x*y + y^2
verbose 1 (my_interreduce_groebner_basis) Normalizing:
verbose 1 (my_interreduce_groebner_basis) Reducing each element with respect to the rest:
verbose 1 (my_reduce) Reducing y^4 + x^2*y + y^3 - x*y with respect to [x*y + y^2]
verbose 1 (my_reduce) Term y^4 cannot be eliminated
verbose 1 (my_reduce) Subtracting x * (x*y + y^2)
verbose 1 (my_reduce) Subtracting -y * (x*y + y^2)
verbose 1 (my_reduce) Term 2*y^3 cannot be eliminated
verbose 1 (my_reduce) Subtracting -1 * (x*y + y^2)
verbose 1 (my_reduce) Term y^2 cannot be eliminated
verbose 1 (my_reduce) Reducing x*y + y^2 with respect to [y^4 + 2*y^3 + y^2]
verbose 1 (my_reduce) Term x*y cannot be eliminated
verbose 1 (my_reduce) Term y^2 cannot be eliminated
[y^4 + 2*y^3 + y^2, x*y + y^2]
sage: set_verbose(0)
sage: R.ideal(G).interreduced_basis()
[y^4 + 2*y^3 + y^2, x*y + y^2]


It seems like there is no such option. However we can program it ourselves:

from sage.misc.verbose import verbose

def my_reduce(f, G):
verbose('Reducing {} with respect to {}'.format(f, G), level=1)
p = f
r = 0
while not p.is_zero():
lt = p.lt()
candidate = next((g for g in G if g.lt().divides(lt)), None)
if candidate is not None:
coeff = lt // candidate.lt()
verbose('Subtracting {} * ({})'.format(coeff, candidate), level=1)
p -= coeff*candidate
else:
verbose('Term {} cannot be eliminated'.format(p.lt()), level=1)
r += p.lt()
p -= p.lt()
return r

def my_interreduce_groebner_basis(G):
verbose('Interreducing my_reduce_groebner_basis(G):
verbose('Reducing the Groebner basis {}'.format(G), level=1)
i = 0
while i < len(G):
j = 0
while j < len(G):
if i != j and G[i].lt().divides(G[j].lt()):
verbose('Removing {} since its leading term is divisible by that of {}'.format(G[j],G[i]), level=1)
G.pop(j)
else:
j += 1
i += 1
verbose('Normalizing:', level=1)
for i in range(len(G)):
if G[i].lc() != 1:
verbose('Dividing {} by {}'.format(G[i], G[i].lc()), level=1)
G[i] = G[i] / G[i].lc()
verbose('Reducing each element with respect to the rest:', level=1)
for i in range(len(G)):
G[i] = my_reduce(G[i], [G[j] for j in range(len(G)) if j != i])
return G


Example:

sage: R.<x,y> = PolynomialRing(QQ,order='deglex')
sage: G = [y^4 + x^2*y + y^3 - x*y, x*y + y^2, x*y^2 + y^3]
sage: R.ideal(G).basis_is_groebner()
True
sage: set_verbose(1)
sage: my_interreduce_groebner_basis(G)
verbose 1 (my_interreduce_groebner_basis) Interreducing my_reduce_groebner_basis(G)
verbose 1 (my_reduce_groebner_basis) Reducing the Groebner basis [y^4 + x^2*y + y^3 - x*y, x*y + y^2, x*y^2 + y^3]
verbose 1 (my_interreduce_groebner_basis) (my_reduce_groebner_basis) Removing x*y^2 + y^3 since its leading term is divisible by that of x*y + y^2
verbose 1 (my_interreduce_groebner_basis) (my_reduce_groebner_basis) Normalizing:
verbose 1 (my_interreduce_groebner_basis) (my_reduce_groebner_basis) Reducing each element with respect to the rest:
verbose 1 (my_reduce) Reducing y^4 + x^2*y + y^3 - x*y with respect to [x*y + y^2]
verbose 1 (my_reduce) Term y^4 cannot be eliminated
verbose 1 (my_reduce) Subtracting x * (x*y + y^2)
verbose 1 (my_reduce) Subtracting -y * (x*y + y^2)
verbose 1 (my_reduce) Term 2*y^3 cannot be eliminated
verbose 1 (my_reduce) Subtracting -1 * (x*y + y^2)
verbose 1 (my_reduce) Term y^2 cannot be eliminated
verbose 1 (my_reduce) Reducing x*y + y^2 with respect to [y^4 + 2*y^3 + y^2]
verbose 1 (my_reduce) Term x*y cannot be eliminated
verbose 1 (my_reduce) Term y^2 cannot be eliminated
[y^4 + 2*y^3 + y^2, x*y + y^2]
sage: set_verbose(0)
sage: R.ideal(G).interreduced_basis()
R.ideal(G).groebner_basis()
[y^4 + 2*y^3 + y^2, x*y + y^2]