Related papers: Non-archimedean Yomdin-Gromov parametrizations and…

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…

We prove an effective version of the Pila-Wilkie Theorem for sets definable using Pfaffian functions, providing effective estimates for the number of algebraic points of bounded height and degree lying on such sets. We also prove effective…

Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the…

We prove that a (globally) subanalytic p-adic function which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken…

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…

We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical…

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

Following the analysis of differentiable mappings of Y. Yomdin, M. Gromov has stated a very elegant "Algebraic Lemma" which says that the "differentiable size" of an algebraic subset may be bounded in terms only of its dimension, degree and…

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of…

In this paper, we generalize results on Zhang's semipositive model metrics from the algebraic setting to strictly analytic spaces over a non-trivially valued non-Archimedean field. We prove stability under pointwise limits and under forming…

Let $K$ be an algebraically closed non-Archimedean field. Leonard Lipshitz has introduced a manageable notion of subanalytic sets of the unit polydisc. This class contains the class of affinoid sets and is stable under projection. We…

We establish a rigid-analytic analog of the Pila-Wilkie counting theorem, giving sub-polynomial upper bounds for the number of rational points in the transcendental part of a $\mathbb{Q}_p$-analytic set, and the number of rational functions…

We describe a class of sharply o-minimal structures, called analytically generated structures, whose definable sets and their complexity filtration are determined by the collection of definable complex cells. We prove a polynomially…

Given an ample line bundle $L$ over a projective $\mathbb{K}$-variety $X$, with $\mathbb{K}$ a non-Archimedean field, we study limits of non-Archimedean metrics on $L$ associated to submultiplicative sequences of norms on the graded pieces…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real valued map on the $n$-torus admits a fibre whose homological size is bounded below by some…

We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale cohomological dimension is…

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of…

We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly…