On a Newton filtration for functions on a curve singularity

Abstract

In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincar\'e series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincar\'e series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincar\'e series only depends on the Newton diagram and not on the defining equation.