ИСТИНА

Facets of a Newton diagram $\Gamma$ in $R^n$ define natural (quasi-homogeneous) valuations on the ring of germs of functions in $n$ variables. Let $f$ is a function with the Newton diagram $\Gamma$. The mentioned valuations define integer-valued functions on the ring of germs of functions on the hypersurface singularity $\{f=0\}$ in two ways. These functions are not, in general, valuations, but order functions. (The latter means that they do not have to possess the property $v(g_1g_2)=v(g_1)+v(g_2)$.) We shall discuss multi-index filtrations defined by these order functions and their Poincar\'e series. The talk reflects joint results with W.Ebeling.