Multivariate Igusa theory: Decay rates of exponential sums

We obtain general estimates for exponential integrals of the form $$E_f(y)=\int_{\mathbb{Z}_{p}^{n}}\psi(\sum_{j=1}^r y_j f_j(x))|dx|,$$ where the $f_j$ are restricted power series over $\mathbb{Q}_p$, $y_j\in\mathbb{Q}_p$, and $\psi$ a nontrivial additive character on $\mathbb{Q}_p$. We prove that if $(f_1,...,f_r)$ is a dominant map, then $|E_f(y)| < c|y|^{\alpha}$ for some $c>0$ and $\alpha<0$, uniform in $y$, where $|y|=\max(|y_i|)_i$. In fact, we obtain similar estimates for a much bigger class of exponential integrals. To prove these estimates we introduce a new method to study exponential sums, namely, we use the theory of $p$-adic subanalytic sets and $p$-adic integration techniques based on $p$-adic cell decomposition. We compare our results to some elementarily obtained explicit bounds for $E_f$ with $f_j$ polynomials.Comment: Improved results and presentatio

Model theory of valued fields

We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.Comment: To appear in Proceedings of Ravello model theory conferenc

Grothendieck rings of Laurent series fields

We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields, for example, fields of Laurent series over the real numbers, over p-adic numbers and over finite fields, and construct definable bijections from the line to the line minus one point.Comment: 9 page

Analytic p-adic Cell Decomposition and Integrals

We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.Comment: The paper is accepted for publication in TAMS. Definitions 4.1 and 4.3 changed, making the integral operator $I$ zero if the parametrized function is not integrable for all parameters, where as before it was zero only in those parameters where the function is not integrabl

Analytic van der Corput Lemma for p-adic and F_q((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems

We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma, where the real condition of sufficient smoothness for the phase is replaced by the condition that the phase is a convergent power series. This van der Corput style result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (analytic) manifolds and opens the way for further applications. As one such further application we give the restriction theorem for Fourier transforms of L^p functions to suitably curved analytic manifolds over non-archimedean local fields, similar to the real restriction result by E. Stein and C. Fefferman

Classification of semi-algebraic pp-adic sets up to semi-algebraic bijection

We prove that two infinite p-adic semi-algebraic sets are isomorphic (i.e. there exists a semi-algebraic bijection between them) if and only if they have the same dimension.Comment: 9 page

Presburger sets and p-minimal fields

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.Comment: to appear in the Journal of Symbolic Logi

Definable sets up to definable bijections in Presburger groups

We entirely classify definable sets up to definable bijections in $\mathbb{Z}$-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets

A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps

A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy, to Kirszbraun's Theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^n\to \mathbb{R}^\ell$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the $p$-adic context that $\widetilde f$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, Lipschitz retractions of $\mathbb{Q}_p^n$ to the topological closure of $X$ when $X$ is definable

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

It was already known that a p-adic, locally Lipschitz continuous semi-algebraic function is piecewise Lipschitz continuous, where 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. 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 parametrized versions and in the subanalytic setting.Comment: 12 page