# Milnor numbers and mixed multiplicities

section 6 of the link Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration $\{m^rJ^s\}$ where J is the Jacobian ideal of the defining polynomial.

Could anyone please illustrate it with some example.

Let $f(x,y,z)=0$ be the equation of a surface with isolated silgularity at the origin. Assume that the coordinates are chosen generically. Then in addition to the Milnor number the mixed multiplicities are the multiplicity of the ideal (x,y,$\partial f/\partial z$) and $(x,\partial f/\partial y,\partial f/\partial z)$. The first one is the order at the origin of a generic partial, which is the order of $f(x,y,z)$ minus one, and is the Milnor number of the intersection of the surface with a general line. By the idealistic Bertini theorem, the second one is the multiplicity of the jacobian ideal of $f(0,y,z)$ and thus the Milbor number of a general hyperplane section of the surface. If you take three integers a>b>c you are in this situation with the Pham-Brieskorn polynomial $x^a+y^b+z^c$.