Hensel minimality

Abstract

We present Hensel minimality, a new notion for non-archimedean tame geometry in Henselian valued fields. This notion resembles o-minimality for the field of reals, both in the way it is defined (though extra care for parameters of unary definable sets is needed) and in its consequences. In particular, it implies many geometric results that were previously known only under stronger assumptions like analyticity. As an application we show that Hensel minimality implies the existence of t-stratifications, as defined previously by the second author. Moreover, we obtain Taylor approximation results which lay the ground for analogues of point counting results by Pila and Wilkie, for analogues of Yomdin's $C^r$-parameterizations of definable sets, and for $p$-adic and motivic integration.Comment: 90 page