milnor and tjurina numbers

Bonjour,


can you me say why the procedure milnor and tjurina gives the same numbers for

{x^{2} + y^{3} + z^{3} +zy=0} or

{x^{2} + y^{3} + y^{4} +yz^{2}=0} ?

thanks.
kaddar.

Please provide the full Singular input for your examples.

I recomputed the examples:
LIB"sing.lib";
ring r = 0,(x,y,z),ds;
poly f = x2 + y3 + z3 +zy;
poly g = x2 + y3 + y4 +yz2;
milnor(f); tjurina(f); //1
milnor(g); tjurina(g); //4

The answer is correct since f is contained in the ideal of the partials.
f is an A_1 and g a D_4 singularity. You get this by applying classify
from classify.lib.

I think it is worth mentioning that by a theorem of Saito any weighted homogeneous polynomial defining an IHS belongs to its Jacobi ideal and viceversa, therefore they have the same Turina and Milnor numbers.