Motivic local density

Abstract

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.

A motivic local Cauchy-Crofton formula

“…Because π −1 (π(x 0 )) ∩X = {x 0 }, the only limit value of c(y) when y goes to π(x 0 ) is x 0 . Since c is ξ -definable and bounded, by Lemma 2.20 of [12] there is an r (ξ ) such that c (B(π(x 0…”

“…We assume the reader is familiar with the notion of motivic local density developed by the author in [12] and in particular with Cluckers and Loeser's theory of motivic integration [5,6]. See [12,Section 2.7] for a short summary of the theory.…”

“…We adopt the notations and conventions of [12], let us recall the main notions we need. We fix T , a tame or mixed-tame theory of valued fields in the sense of [6].…”

“…A notion of local density for definable sets in Henselian valued fields of characteristic zero has been developed by the author in [12]. The aim of this note is to establish a motivic analogue of the local Cauchy-Crofton formula.…”

Some results of algebraic geometry over Henselian rank one valued fields

“…4]). Yet another approach for the case of tame theories is provided in [18,Lemma 2.20]. The second conclusion relies on Puiseux's parametrization.…”