D.5.4.1 gmsring

Procedure from library gaussman.lib (see gaussman_lib).

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 gmsmaxdeg; maximal weight of variables

Note:

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

Example:

LIB "gaussman.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,
→ 2xy2+5x4,
→ 5x5-5y5,
→ 10y6+25x3y4
print(gmsmatrix);
→ 0,1,x, -2xy,  
→ 1,0,-y,2y2+5x3
print(gmsbasis);
→ y5,
→ y4,
→ y3,
→ y2,
→ xy,
→ y,
→ x4,
→ x3,
→ x2,
→ x,
→ 1
gmsmaxdeg;
→ 1