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Hilbert Series for S=Q[x,y,z]/I when I has parameter
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2467		 Page 1 of 1
Author:  gstic [ Wed Nov 05, 2014 12:15 am ]
Post subject:  Hilbert Series for S=Q[x,y,z]/I when I has parameter
Hello,
I have a problem when I try to compute Hilbert Series for S=Q[x,y,z]/I when I is an homogeneous ideal with parameter

Let f = x^3+y^3+z^3-3*a*x*y*z be a polynomial.
Is known that :
-for a^3=1, {f=0} is singular and Hilbert Series is infinite
-for a^3<>1 {f=0} is non-singular (smooth) and Hilbert Series is finite
Singular package can work with parameters but does not discriminate in this case...

The question is Why does not work the following program, without fix the parameter with minpoly
If we do not fix the parameter,
ring R=(0,a),(x,y,z),dp;
poly f;
f=x^3+y^3+z^3-3*a*x*y*z;
ideal G=std(jacob(f));
hilb(G,2);
//--> Hilbert series for S is H(t)= 1+3t+3t^2+t^3 (polynomial)
the result is wrong for some values (a^-1=0)....

If we fix the parameter with minpoly, we get the correct result:
ring R=(0,a),(x,y,z),dp;
minpoly =a^3-1; // or minpoly=a-1 or minpoly=a^2+a+1
poly f;
f=x^3+y^3+z^3-3*a*x*y*z;
ideal G=std(jacob(f));
hilb(G,2);
//--> Hilbert series for S is H(t)=
(1+2t)/(1-t)=1+3t+3t^2+3t^3+3t^4+... (infinite)

Without parameter, everything is OK:
for a = 1
ring R=0,(x,y,z),dp;
poly f;
int a; a=1;
f=x^3+y^3+z^3-3*a*x*y*z;
ideal G=std(jacob(f));
hilb(G,2);
//--> Hilbert series for S is H(t)=
(1+2t)/(1-t)=1+3t+3t^2+3t^3+3t^4+... (infinite)
Also for a^3<>1 eg a= 2 we get the correct result.
//--> Hilbert series for S is H(t)= 1+3t+3t^2+t^3 (polynomial)

Is possible to work with parameters without fix them to compute Hilbert Series?


Thank's in advance
Author:  hannes [ Fri Nov 07, 2014 11:46 am ]
Post subject:  Re: Hilbert Series for S=Q[x,y,z]/I when I has parameter
For the computation of the Hilbert Series only the leading monomials
(power product without coefficients) of the standard basis are considered: i.e. every
expression in a is considered to be non-zero if not identical to 0.
If you need to distinguish the different cases for different values of a
you have to compute the different standard bases first:
see for example grobcov.lib http://www.singular.uni-kl.de/Manual/4-0-1/sing_953.htm
Author:  gstic [ Wed Nov 12, 2014 11:25 pm ]
Post subject:  Re: Hilbert Series for S=Q[x,y,z]/I when I has parameter
Thanks you very much for the answer.
Now it works !
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