Lojasiewicz exponents, the integral closure of ideals and Newton polyhedra

Abstract

We give an upper estimate for the \L ojasiewicz exponent l(J, I) of an ideal J⊂eq A(\bm{K}ⁿ) with respect to another ideal I in the ring A(\bm{K}ⁿ) of germs analytic functions f:(\bm{K}ⁿ, 0)→ K, where \bm{K}=\bm{C} or \bm{R}, using Newton polyhedrons. In particular, we give a method to estimate the \L ojasiewicz exponent α₀(f) of a germ f∈ A(\bm{K}ⁿ) that can be applied when f is Newton degenerate with respect to its Newton polyhedron.