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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