A Method to Estimate the Degree of C0-Sufficiency of Analytic Functions

The problem of determining the degree of C-sufficiency s(f) of a given analytic function germ f : (C, 0) → (C, 0) is well known in singularity theory. There is some previous work which gives an upper estimate for this number (see Section 2 for its definition). For instance, in the paper [Lichtin 81], the case n = 2 is considered. In the paper [Fukui 91], an estimate for the degree of C-sufficiency of Newton non-degenerate function germs (in the sense of [Kouchnirenko 76]) is obtained for any n, thus generalizing the cited work of Lichtin. In this paper, we use some facts from commutative algebra, particularly from multiplicity theory, to give a method providing an estimation for s(f), where f : (C, 0) → (C, 0) is any analytic function germ with an isolated singularity at the origin. The number thus obtained differs from s(f) at most by one unit. As we shall see, the characterization of s(f) using à Lojasiewicz type inequalities given by the results of [Chang and Lu 73] and [Bochnak and Kucharz 79] will play a fundamental role in our approach. The link between the algebraic tools we use and the language of à Lojasiewicz type inequalities comes from [Lejeune and Teissier 74] characterizing the integral closure of an ideal in the ring On of analytic function germs (C, 0) → C (see Theorem 2.4).