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Abstract

We give criteria of finite determinacy for the volume and multiplicities. Given an analytic set described by $\{ v=0\} $, we prove that the log-analytic expansion of the volume of the intersection of the set and a “little ball” is determined by that of the set defined by the Taylor expansion of $v$ up to a certain order if the mapping $v$ has an isolated singularity at the origin. We also compare the cardinalities of finite fibers of projections restricted to such a set.