Łojasiewicz exponents, the integral closure of ideals and Newton polyhedra

Abstract

We give an upper estimate for the Łojasiewicz exponent $\ell (J,I)$ of an ideal $J\subseteq A\left(K^n\right)$ with respect to another ideal I in the ring $A\left(K^n\right)$ of germs analytic functions $f$ : $(K^n,0)\to K$, where $K=C$ or $R$, using Newton polyhedrons. In particular, we give a method to estimate the Łojasiewicz exponent ${\alpha }_0\left(f\right)$ of a germ $f\in A\left(K^n\right)$ that can be applied when $f$ is Newton degenerate with respect to its Newton polyhedron.