Singular

7.7.7.0. lpDHilbertSickle

Usage:

lpDHilbertSickle(G[,degbound,n]); G an ideal, degbound, n optional
integers

Return:

list

Purpose:

Computing K-dimension, Hilbert series and mistletoes at once

Assume:

- basering is a Letterplace ring.
- if you specify a different degree bound degbound,
degbound <= attrib(basering,uptodeg) holds.

Note:

- If L is the list returned, then L[1] is an integer, the K-dimension,
L[2] is an intvec, the Hilbert series and L[3] is an ideal,
the mistletoes
- If degbound is set, there will be a degree bound added. 0 means no
degree bound. Default: attrib(basering,uptodeg).
- n can be set to a different number of variables.
Default: n = attrib(basering, lV).
- If I = L[1] is the intvec returned, then I[k] is the (k-1)-th
coefficient of the Hilbert series.
- If the K-dimension is known to be infinite, a degree bound is needed

LIB "fpadim.lib";
ring r = 0,(x,y),dp;
def R = makeLetterplaceRing(5); // constructs a Letterplace ring
setring R; // sets basering to Letterplace ring
ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a
//Groebner basis
lpDHilbertSickle(G,5,2); //invokes procedure with degree bound 5 and 2 variables
==> [1]:
==>    6
==> [2]:
==>    1,2,2,1
==> [3]:
==>    _[1]=x(1)*y(2)
==>    _[2]=y(1)*x(2)*y(3)
// note that the optional parameters are not necessary, due to the finiteness
// of the K-dimension of the factor algebra
lpDHilbertSickle(G); // procedure with ring parameters
==> [1]:
==>    6
==> [2]:
==>    1,2,2,1
==> [3]:
==>    _[1]=x(1)*y(2)
==>    _[2]=y(1)*x(2)*y(3)
lpDHilbertSickle(G,0); // procedure without degreebound
==> [1]:
==>    6
==> [2]:
==>    1,2,2,1
==> [3]:
==>    _[1]=x(1)*y(2)
==>    _[2]=y(1)*x(2)*y(3)