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Motivic local density

Authors :
Forey, Arthur
Publication Year :
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.<br />Comment: 40 pages, minor corrections

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1512.00420
Document Type :
Working Paper
Full Text :
https://doi.org/10.1007/s00209-016-1829-0