arXiv subjects
Publications and source records attributed to Immanuel Halupczok.
We study the ring of Puiseux polynomials with integer coefficients. We prove notably that the order given by the leading coefficient is definable without parameters in the language of rings. This answers a question of R. Robinson.
We correct the statements and proofs of the (auxiliary) Propositions 4.1 and 4.2 of our paper `Evaluation of motivic functions, non-nullity, and integrability in fibers' in Advances in Mathematics, Vol. 409, Part A, Paper No. 108635, 29 pages (2022), and we explain how the proofs of the main results can be adapted to work with those corrected propositions.
We prove that non-abelian definable, definably simple groups in 1-h-minimal henselian valued fields are essentially already linear algebraic groups. Here, the group is assumed to live in the home sort. We have a similar result in pure algebraically closed valued fields of positive characteristic, under the additional assumption that the definable group is a subgroup of a linear algebraic group.
We prove the nonarchimedean counterpart of a real inequality involving the metric entropy and measure geometric invariants $V_i$, called Vitushkin's variations. Our inequality is based on a new convenient partial preorder on the set of constructible motivic functions, extending the one considered by R. Cluckers and F. Loeser in Constructible motivic functions and motivic integration, Invent. Math., 173 (2008). We introduce, using motivic integration theory and the notion of riso-triviality, nonarchimedean substitutes of the Vitushkin variations $V_i$, and in particular of the number $V_0$ of connected components. We also prove the nonarchimedean global Cauchy-Crofton formula for definable sets of dimension $d$, relating $V_d$ and the motivic measure in dimension $d$.
If a reduced bivariate polynomial is quasi-homogeneous, then its discriminant is a monomial. Over fields of characteristic $0$, we show that if one adds another simple condition, this becomes an equivalence. We also give a third equivalent condition that is stated geometrically.
These are notes from a mini-course about the main results of arXiv:2206.03438: I explain how, using suitable valued fields, one obtains a natural notion of canonical stratifications (of e.g. algebraic subsets of $\mathbb{R}^n$). I also explain how the same techniques yield more invariants of singularities, and I present an application to Poincaré series. While some rudimentary knowledge of model theory is useful, the notes should also be accessible without such knowledge. In particular, they contain an introduction to the non-standard analysis needed for this approach.
We introduce a new notion of stratification (``riso-stratification''), which is canonical and which exists in a variety of settings, including different topological fields like $\mathbb{C}$, $\mathbb{R}$ and $\mathbb{Q}_p$, and also including different o-minimal structures on $\mathbb{R}$. Riso-stratifications are defined directly in terms of a suitable notion of triviality along strata; the key difficulty and main result is that the strata defined in this way are ``algebraic in nature'', i.e., definable in the corresponding first-order language. As an example application, we show that local motivic Poincaré series are, in some sense, trivial along the strata of the riso-stratification. Behind the notion of riso-stratification lies a new invariant of singularities, which we call the ``riso-tree'', and which captures, in a canonical way, information that was contained in the non-canonical strata of a Lipschitz stratification. On our way to the Poincaré series application, we show, among others, that our notions interact well with motivic integration.
In this paper together with the preceding Part I \cite{CHR}, we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to \cite{CHR} the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality, more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.
We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is $0$-hmix-minimality (which, in equi-characteristic $0$, amounts to $0$-h-minimality).
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.
We define an operation of evaluation at a point for motivic constructible (exponential) functions from the Cluckers-Loeser framework of motivic integration and show that two such motivic functions are abstractly equal if and only if their evaluations at each point are the same. We similarly characterise relative integrability in terms of integrability in each fiber separately. These results simplify the mentioned frameworks of motivic integration and their usage.
We introduce two new notions of stratifications in valued fields: t$^2$-stratifications and arc-wise analytic t-stratifications. We show the existence of arc-wise analytic t-stratifications in algebraically closed valued fields with analytic structure in the sense of R. Cluckers and L. Lipshitz. We prove that arc-wise analytic t-stratifications are t$^2$-stratifications and, moreover, that t$^2$-stratifications are valuative Lipschitz stratifications as defined by the second author and Y. Yin (the latter ones being closely related to Lipschitz stratifications in the sense of Mostowski). Finally, we introduce a combinatorial invariant associated to a t-stratification which we call the critical value function. We explain how the critical value function of arc-wise analytic t-stratifications can be used to formulate programatic conjectural bounds for the Nash-Semple conjecture.
We prove that if two semi-algebraic subsets of $\mathbb{Q}_p^n$ have the same $p$-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a $p$-adic analogue of a question asked by Kontsevich-Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over $\mathbb{Q}_p$, universal motivic integration (in the sense of Hrushovski-Kazhdan) is just $p$-adic integration.
We study upper bounds, approximations, and limits for functions of motivic exponential class, uniformly in non-Archimedean local fields whose characteristic is $0$ or sufficiently large. Our results together form a flexible framework for doing analysis over local fields in a field-independent way. As corollaries, we obtain many new transfer principles, for example, for local constancy, continuity, and existence of various kinds of limits. Moreover, we show that the Fourier transform of an $L^2$-function of motivic exponential class is again of motivic exponential class. As an application in the realm of representation theory, we prove uniform bounds for the normalized by the discriminant Fourier transforms of orbital integrals on connected reductive $p$-adic groups.
We study some constructions on distributions in a uniform $p$-adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of ${\mathscr C}^{\mathrm{exp}}$-class and which is based on the notion of ${\mathscr C}^{\mathrm{exp}}$-class functions from [6]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non-archimedean local fields. We study wave front sets, pull-backs and push-forwards of distributions of this class. In particular we show that the wave front set is always equal to the complement of the zero locus of a ${\mathscr C}^{\mathrm{exp}}$-class function. We first revise and generalize some of the results of Heifetz that he developed in the $p$-adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz-Bruhat functions and their push forwards in relation to discriminants.
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.
These are notes of a series of talks about motivic integration I gave on the Münster Model Theory Month. Readers are assumed to have some basic knowledge of model theory and of valued fields. The notes are closest to the Cluckers-Loeser style of motivic integration, though mostly they are about doing p-adic integration uniformly in all $\mathbb{Q}_p$.
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 previous papers. 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.