It looks as if Sage blindly runs is_homogeneous() - and this is not needed.
Namely, after I get rid of these checks:
-- a/src/sage/rings/polynomial/multi_polynomial_ideal.py
+++ b/src/sage/rings/polynomial/multi_polynomial_ideal.py
@@ -3025,8 +3025,8 @@ class MPolynomialIdeal_singular_repr(
sage: I.hilbert_series()
# optional -
sage.rings.number_field
(t^4 + t^3 + t^2 + t + 1)/(t^2 - 2*t + 1)
"""
- if not self.is_homogeneous():
- raise TypeError("ideal must be homogeneous")
+# if not self.is_homogeneous():
+# raise TypeError("ideal must be homogeneous")
if algorithm == 'sage':
from sage.rings.polynomial.hilbert import hilbert_poincare_series
@@ -3119,8 +3119,8 @@ class MPolynomialIdeal_singular_repr(
True
"""
- if not self.is_homogeneous():
- raise TypeError("Ideal must be homogeneous.")
+# if not self.is_homogeneous():
+# raise TypeError("Ideal must be homogeneous.")
if algorithm == 'sage':
from sage.rings.polynomial.hilbert import first_hilbert_series
then at least I'm able to compute the numerator of the series, and the
series itself, correctly:
sage: T.<a,b>=QQ[]
sage: I=T.ideal(a^2-b^3);
sage: I.hilbert_numerator(grading=[3,2])
-t^6 + 1
sage: I.hilbert_series(grading=[3,2])
(t^2 - t + 1)/(-t + 1)
(the latter looks wrong, but it's just the correct expression simplified
sage: (1-t^6)/((1-t^3)*(1-t^2))
(-t^2 + t - 1)/(t - 1)
On Thu, Jul 13, 2023 at 2:45 PM Dima Pasechnik <dimp...@gmail.com> wrote:
>
> On Tue, Jul 4, 2023 at 12:26 PM Kwankyu Lee <ekwan...@gmail.com> wrote:
> >
> > Also, as far as I understand, Sage can compute the minimal free resolution
> > of
> > the module of syzygies of S, and from the resolution the presentation can be
> > assembled.
> >
> >
> > Yes. It's here:
> > https://doc.sagemath.org/html/en/reference/resolutions/index.html
> >
> > So it seems that the only missing bit is computation of a presentation of S.
> >
> >
> > Let phi: R[y_1,...,y_k] -> R[x_1,...,x_n] mapping by y_i -> f_i. Then,
> > perhaps, your I is the kernel of phi.
>
> in default grading, the generators of the kernel need not be homogeneous.
> You need to grade y_i with degrees of f_i, for this to work correctly,
> but Sage's PolynomialRing does not have such an
> option, even though it is available in Singular (and in Macaulay2).
>
> E.g. in Singular:
> $ ./sage --singular
> ...
> > ring T = 0, (a,b), dp;
> > ideal I = a^2-b^3; // a toy example
> > intvec iv = [3,2]; // degrees for a and b
> > hilb(I,1,iv);
> 1,0,0,0,0,0,-1,0
>
> but in Sage:
>
> sage: T.<a,b>=QQ[]
> sage: I=T.ideal(a^2-b^3);
> sage: I.hilbert_series(grading=(3,2))
> ---------------------------------------------------------------------------
> TypeError Traceback (most recent call last)
> Cell In [5], line 1
> ----> 1 I.hilbert_series(grading=(Integer(3),Integer(2)))
>
> File
> /mnt/opt/Sage/sage-dev/src/sage/rings/polynomial/multi_polynomial_ideal.py:300,
> in RequireField.__call__(self, *args, **kwds)
> 298 if not R.base_ring().is_field():
> 299 raise ValueError("Coefficient ring must be a field for
> function '%s'."%(self.f.__name__))
> --> 300 return self.f(self._instance, *args, **kwds)
>
> File /mnt/opt/Sage/sage-dev/src/sage/rings/qqbar_decorators.py:96, in
> handle_AA_and_QQbar.<locals>.wrapper(*args, **kwds)
> 90 from sage.rings.abc import AlgebraicField_common
> 92 if not any(isinstance(a, (Polynomial, MPolynomial, Ideal_generic))
> 93 and isinstance(a.base_ring(), AlgebraicField_common)
> 94 or is_PolynomialSequence(a)
> 95 and isinstance(a.ring().base_ring(),
> AlgebraicField_common) for a in args):
> ---> 96 return func(*args, **kwds)
> 98 polynomials = []
> 100 for a in flatten(args, ltypes=(list, tuple, set)):
>
> File
> /mnt/opt/Sage/sage-dev/src/sage/rings/polynomial/multi_polynomial_ideal.py:3029,
> in MPolynomialIdeal_singular_repr.hilbert_series(self, grading,
> algorithm)
> 2955 r"""
> 2956 Return the Hilbert series of this ideal.
> 2957
> (...)
> 3026 (t^4 + t^3 + t^2 + t + 1)/(t^2 - 2*t + 1)
> 3027 """
> 3028 if not self.is_homogeneous():
> -> 3029 raise TypeError("ideal must be homogeneous")
> 3031 if algorithm == 'sage':
> 3032 from sage.rings.polynomial.hilbert import hilbert_poincare_series
>
> TypeError: ideal must be homogeneous
>
>
>
>
>
>
>
> >
> >
> > --
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