Singular

D.6.7.1 gmsring

Usage:

gmsring(t,s); poly t, string s

Assume:

characteristic 0; local degree ordering;
isolated critical point 0 of t

Return:

ring G; Gauss-Manin system of t with variable s poly gmspoly=t; ideal gmsjacob; Jacobian ideal of t ideal gmsstd; standard basis of Jacobian ideal matrix gmsmatrix; matrix(gmsjacob)*gmsmatrix==matrix(gmsstd) ideal gmsbasis; monomial vector space basis of Jacobian algebra int Gmssing::gmsmaxdeg; maximal weight of variables

Note:

gmsbasis is a C[[s]]-basis of H" and [t,s]=s^2

LIB "gmssing.lib";
ring @R=0,(x,y),ds;
poly t=x5+x2y2+y5;
def G=gmsring(t,"s");
setring(G);
gmspoly;
==> x2y2+x5+y5
print(gmsjacob);
==> 2xy2+5x4,
==> 2x2y+5y4
print(gmsstd);
==> 2x2y+5y4,
==> 5x5-5y5,
==> 2xy2+5x4,
==> 10y6+25x3y4
print(gmsmatrix);
==> 0,x, 1,-2xy,  
==> 1,-y,0,2y2+5x3
print(gmsbasis);
==> y5,
==> y4,
==> y3,
==> y2,
==> xy,
==> y,
==> x4,
==> x3,
==> x2,
==> x,
==> 1
Gmssing::gmsmaxdeg;
==> 1