We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F q [t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.

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 construct a new motivic integration morphism, the so-call bounded integral, that interpolates both the integration morphisms with and without volume forms of Hrushovski and Kazhdan. This is done within the framework of model theory of algebraically closed valued fields of equicharacteristic zero. As an application, we recover and extend some results of Hrushovski and Loeser about the motivic Milnor fiber.

Let k be a field of characteristic zero containing all roots of unity and K = k((t)). We build a ring morphism from the Grothendieck group of semi-algebraic sets over K to the Grothendieck group of motives of rigid analytic varieties over K. It extend the morphism sending the class of an algebraic variety over K to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over K and quasi-unipotent motives over k ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.

We introduce a notion of complexity of a complex of ℓ \ell -adic sheaves on a quasi-projective variety and prove that the six operations are “continuous”, in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields.