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We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this first paper we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application., Comment: 90 pages; changes compared to v2: proof of dimension theory is more self-contained; new section 5.8 about motivic integration

We propose to grok Lipschitz stratifications from a non-archimedean point of view and thereby show that they exist for closed definable sets in any power-bounded o-minimal structure on a real closed field. Unlike the previous approaches in the literature, our method bypasses resolution of singularities and Weierstrass preparation altogether; it transfers the situation to a non-archimedean model, where the quantitative estimates appearing in Lipschitz stratifications are sharpened into valuation-theoretic inequalities. Applied to a uniform family of sets, this approach automatically yields a family of stratifications which satisfy the Lipschitz conditions in a uniform way., Comment: 44 pages, 5 figures

We prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

We give an example of a parameter-free definable Henselian valuation ring which is neither definable by a parameter-free inline image-formula nor by a parameter-free inline image-formula in the language of rings. This answers a question of Prestel.

We provide a short and self-contained overview of the techniques based on motivic integration as they are applied in harmonic analysis on p-adic groups; our target audience is mainly representation theorists with no back- ground in model theory (and model theorists with an interest in recent ap- plications of motivic integration in representation theory, though we do not provide any representation theory background). We aim to give a fairly com- prehensive survey of the results in harmonic analysis that were proved by such techniques in the last ten years, with emphasis on the most recent techniques and applications from (13), (8), and (32, Appendix B).

Let $G$ be a connected reductive algebraic group over a non-Archimedean local field $K$, and let $g$ be its Lie algebra. By a theorem of Harish-Chandra, if $K$ has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in $g(K)$ by locally constant functions, which, extended by zero to all of $g(K)$, are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer principles for integrability and boundedness conditions for motivic exponential functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem holds also when $K$ is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis on the existence of the mock exponential map, this also implies local integrability of Harish-Chandra characters of admissible representations of $G(K)$, where $K$ is an equicharacteristic field of sufficiently large (depending on the root datum of $G$) characteristic., Comment: Compared to v2/v3: some proofs simplified, the main statement generalized; slightly reorganized. Regarding the automatically generated text overlap note: it overlaps with the Appendix B (which is part of arXiv:1208.1945) written by us; the appendix and this article cross-reference each other, and since the set-up is very similar, some overlap is unavoidable

We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from Cluckers and Loeser (Ann Math 171:1011–1065, 2010) and Shin and Templier (Invent Math, 2015, Appendix B). These functions come from rather general oscillatory integrals on local fields, and can be used to describe, e.g., Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.

It is known that a p-adic, locally Lipschitz continuous semi-algebraic function, is piecewise Lipschitz continuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates. As a corollary, we find that all definable functions in ordered abelian groups are piecewise affine linear on finitely many definable pieces., 30 pages

Let A A be a zero-sum free subset of Z n \mathbb {Z}_n with | A | = k |A|=k . We compute for k ≤ 7 k\leq 7 the least possible size of the set of all subset-sums of A A .

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective., Comment: 15 pages, to appear in JSL; minor corrections and one proof replaced by a sketch of proof and a reference

Due to our lack in higher dimensional imagination, it is difficult to find explicit strategies for higher dimensional animal achievement games. Here, we give two methods to build up strategies step by step for increasing dimension. As applications we obtain improved bounds for the winning dimensions of certain polyominoes and new bounds for hypercube Tic-Tac-Toe with and without diagonals.

In “Pseudo-finite fields and related structures” Hrushovski asks (in a more general context) whether the notion of measure on definable sets in pseudo-finite fields can be extended to perfect PAC fields whose Galois group is bounded but not Z ˆ . We define a suitable generalization of measure and give an answer to this question: yes if the Galois group is pro-cyclic, no otherwise.

© Les auteurs, 2018. Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from [10], [11] and [5]. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-Pas quantifier elimination which allows us to understand definable sets in the value group and in the valued field. ispartof: Journal de l'Ecole Polytechnique - Mathematiques vol:5 pages:45-78 status: published

In the context of the local Langlands philosopy for R \mathbb {R} , Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group G ( R ) G(\mathbb {R}) and the space of “complete geometric parameters”—a space of equivariant local systems on a variety on which the Langlands-dual of G ( R ) G(\mathbb {R}) acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of categories. In this article, that conjecture is proven in the case where G ( R ) G(\mathbb {R}) is a generalized Lorentz group SO n , 1 ⁡ ( R ) \operatorname {SO}_{n,1}(\mathbb {R}) .

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over $Q_p^n$ implies integrability over $F_p ((t))^n$ for large $p$, and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind., Comment: 44 pages

We study the problem of arranging a set of n disks with prescribed radii on n rays emanating from the origin such that two neighboring rays are separated by an angle of 2π/n. The center of the disks have to lie on the rays, and no two disk centers are allowed to lie on the same ray. We require that the disks have disjoint interiors, and that for every ray the segment between the origin and the boundary of its associated disk avoids the interior of the disks. Let $\widetilde r$ be the sum of the disk radii. We introduce a greedy strategy that constructs such a disk arrangement that can be covered with a disk centered at the origin whose radius is at most $2\widetilde r$ , which is best possible. The greedy strategy needs O(n) arithmetic operations. As an application of our result we present an algorithm for embedding unordered trees with straight lines and perfect angular resolution such that it can be covered with a disk of radius n3.0367, while having no edge of length smaller than 1. The tree drawing algorithm is an enhancement of a recent result by Duncan et al. [Symp. of Graph Drawing, 2010] that exploits the heavy-edge tree decomposition technique to construct a drawing of the tree that can be covered with a disk of radius 2 n4.

We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also contain non-local information about the stratified sets. For example, they do not only see the singularities in the valued field, but also those in the residue field. Like Whitney stratifications, t-stratifications exist for different classes of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic subsets. The general framework are definable sets (in the sense of model theory) in a language that satisfies certain hypotheses. We give two applications. First, we show that t-stratifications in suitable valued fields $K$ induce classical Whitney stratifications in $\Bbb R$ or $\Bbb C$; in particular, the existence of t-stratifications implies the existence of Whitney stratifications. This uses methods of non-standard analysis. Second, we show how, using t-stratifications, one can determine the ultra-metric isometry type of definable subsets of $\Bbb Z_p^n$ for $p$ sufficiently big. For those $p$, this proves a conjecture stated in a previous article. In particular, this yields a new, geometric proof of the rationality of Poincar\'e series., Comment: Fixed typos; enhanced the presentation

A fairly long-standing conjecture is that the Davenport constant of a group G D Zn1 ˚ � � � ˚ Znk withn1 j � � � j nk is1 C P kD1 .ni 1/. This conjecture is false in general, but it remains to know for which groups it is true. By using inductive methods we prove that for two fixed integersk andit is possible to decide whether the conjecture is satisfied for all groups of the form Z ` ˚ Zn withn co-prime tok. We also prove the conjecture for groups of the form Z3 ˚ Z3nZ3n; wheren is co-prime to6, assuming a conjecture about the maximal zero-sum free sets in Z 2 .

To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and for one-dimensional sets., Comment: 33 pages, 1 figure

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything., 27 pages, 3 figures