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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.
A stratified form is a collection of forms defined on the strata of a stratification of a subanalytic set and satisfying a continuity property when we pass from one stratum to another. We prove that these forms satisfy Stokes' formula on subanalytic singular simplices.
We associate to a given polynomial map from to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.
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