We now fix an ordering <_{m} on
compatible with < and denote it
also with <. Again we have the notion of coefficient c(f) and leading
monomial L(f). < has the important property:
In SINGULAR submodules of free modules are defined by a set of generators. These sets are of type module.
SINGULAR example (see [PS]):
// ===========Poincare complex ========================== // counterexample to a possible generalization of a theorem of Kyoji // Saito. A complete intersection with exact Poincare complex at 0 // but which is in no coordinate system weighted homogeneous // see [PS] for an exeact decription. // // define (Z/32003)[[x,y,z]] ring Rp=32003,(x,y,z),(c,ds); // load additional procedures (milnor, tjurina) LIB "sing.lib"; // select an example, parametrized by n and m int n=883; int m=937; poly f1=xy+z^(n-1); poly f2=xz+y^(m-1)+yz2; ideal f=f1,f2; // define the basering as Rp/f and fetch the data qring R=std(f); ideal f=fetch(Rp,f); poly f1,f2=fetch(Rp,f1),fetch(Rp,f2); // the module Omega2: module omega2= [diff(f1,y),diff(f1,z),0], [diff(f1,x),0,-diff(f1,z)], [0,diff(f1,x),diff(f1,y)], [diff(f2,y),diff(f2,z),0], [diff(f2,x),0,-diff(f2,z)], [0,diff(f2,x),diff(f2,y)]; //it can be shown, that the Poincare complex is exact, if (in this case) //Milnor number(f)+1 = multiplicity(omega2) omega2=std(omega2); multiplicity(omega2); // The Milnor number of the complete intersection f"; milnor(f); // The Tjurina number of the complete intersection f tjurina(f); //since the Milnor number and the Tjurina number do not coincide, //the singularity is not weighted homogeneous