Submission date: 1 December 2015

Abstract:

We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about p-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition (w_f). Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the theory of the residue sorts. We then prove the existence of a distinguished tangent cone and that one can compute the local density on this cone endowed with appropriate motivic multiplicities. As an application we prove a uniformity theorem for p-adic local density. We also correct some inaccuracies found in [2].

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Full text arXiv 1512.00420: pdf, ps.