Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.09852

From Cell Decomposition to Motivic Integration, Hensel Minimality, and Point Counting

Raf Cluckers
View original source ↗Read source document ↗Export metadata ↓

Abstract

Following Macintyre's quantifier elimination result for semi-algebraic $p$-adic sets from 1976, Denef's cell decomposition from 1984 has played a catalyzing role for both motivic integration and Hensel minimality. Motivic integration has in turn been applied in the Langlands program via transfer principles to change the characteristic of the local field, e.g., for the fundamental lemma. Hensel minimality has been used to study rational points on definable sets, providing non-archimedean analogues of results by Pila and Wilkie in o-minimal structures. I will review some related results and open questions.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Raf Cluckers. 2026-10-07. From Cell Decomposition to Motivic Integration, Hensel Minimality, and Point Counting. https://doi.org/10.1137/25m180545x

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Complexity of deep computations via topology of function spaces

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification of Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.

math.LO↗
arXiv2026

Points and their multiples on curves in powers of simple abelian varieties

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.

math.LO↗
arXiv2026

Condensed Whitehead Problem

The Whitehead problem, which asks whether every abelian group $A$ satisfying $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $ A $ of size $κ$ is free if and only if $\underline{\mathrm{Ext}}^1(\underline{A}, \underline{\mathbb{Z}})(S_κ) = 0$, where $ S_κ$ is the Stone space of the Boolean completion of $\operatorname{Add}(ω,κ)$.

math.LO↗