On Saturday 21 February 2009, davidp wrote:
> Hi,
>
> Singular's hilb command does not work as expected:
>
> sage: R = singular.ring(0,'(x,y,z)','dp')
> sage: I = singular.ideal(['x^3-y^2*z','z^2-x*y'])
> sage: I.hilb()
> `sage90`
>
> Could someone please explain this?
The hilb() command only prints information and returns nothing, which we don't
deal with at this level. You can work around this by:
sage: R = singular.ring(0,'(x,y,z)','dp')
sage: I = singular.ideal(['x^3-y^2*z','z^2-x*y'])
sage: print singular.eval("hilb(%s)"%I.name())
// ** sage43 is no standard basis
// 1 t^0
// -1 t^2
// -1 t^3
// 1 t^4
// 1 t^0
// 1 t^1
// -1 t^3
// dimension (proj.) = 1
// degree (proj.) = 1
or you can use Sage's native commands:
sage: P.<x,y,z> = QQ[]
sage: I = (x^3-y^2*z,z^2-x*y)*P;
sage: I.hilbert_series()
(t^3 + 2*t^2 + 2*t + 1)/(-t + 1)
sage: I.hilbert_series?
Type: instancemethod
Base Class: <type 'instancemethod'>
String Form: <bound method MPolynomialIdeal.hilbert_series of Ideal (x^3 -
y^2*z, -x*y + z^2) of Multivariate Polynomial Ring in x, y, z over Rational
Field>
Namespace: Interactive
File:
/home/malb/SAGE/local/lib/python2.5/site-packages/sage/rings/polynomial/multi_polynomial_ideal.py
Definition: I.hilbert_series(self, singular=Singular)
Docstring:
Return the Hilbert series of this ideal.
Let I = self be a homogeneous ideal and R =
self.ring() be a graded commutative algebra (R =
oplus R_d) over a field K. Then the Hilbert function is
defined as H(d) = dim_K R_d and the Hilbert series of I is
defined as the formal power series HS(t) = sum_0^{infty} H(d) t^d.
This power series can be expressed as HS(t) = Q(t)/(1-t)^n
where Q(t) is a polynomial over Z and n the number of
variables in R. This method returns Q(t)/(1-t)^n.
EXAMPLE:
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
sage: I.hilbert_series()
(-t^4 - t^3 - t^2 - t - 1)/(-t^2 + 2*t - 1)
Cheers,
Martin
--
name: Martin Albrecht
_pgp: http://pgp.mit.edu:11371/pks/lookup?op=get&search=0x8EF0DC99
_otr: 47F43D1A 5D68C36F 468BAEBA 640E8856 D7951CCF
_www: http://www.informatik.uni-bremen.de/~malb
_jab: [email protected]
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