D.5.7.1 ArnoldAction

Procedure from library qhmoduli.lib (see qhmoduli_lib).

Usage:

ArnoldAction(f, [Gf, B]); poly f; list Gf, B;
’Gf’ is a list of two rings (coming from ’StabEqn’)

Purpose:

compute the induced action of the stabilizer G of f on T_, where T_ is given by the upper monomials B of the Milnor algebra of f.

Assume:

f is quasihomogeneous

Return:

polynomial ring over the same ground field, containing the ideals ’actionid’ and ’stabid’.
- ’actionid’ is the ideal defining the induced action of Gf on T_
- ’stabid’ is the ideal of the stabilizer Gf in the new ring

Example:

LIB "qhmoduli.lib";
ring B   = 0,(x,y,z), ls;
poly f = -z5+y5+x2z+x2y;
def R = ArnoldAction(f);
setring R;
actionid;
→ actionid[1]=-s(2)*t(1)+s(3)*t(1)
→ actionid[2]=-s(2)^2*t(2)+2*s(2)^2*t(3)^2+s(3)^2*t(2)
→ actionid[3]=s(2)*t(3)+s(3)*t(3)
stabid;
→ stabid[1]=s(2)*s(3)
→ stabid[2]=s(1)^2*s(2)+s(1)^2*s(3)-1
→ stabid[3]=s(1)^2*s(3)^2-s(3)
→ stabid[4]=s(1)^2+s(2)^4-s(3)^4
→ stabid[5]=s(1)^4+s(2)^3-s(3)^3
→ stabid[6]=-s(1)^2*s(3)+s(3)^5