arXiv subjects

Raf Cluckers

Publications and source records attributed to Raf Cluckers.

At least 19 recordsLinked to original sources

arXiv2026

Corrigendum to `Evaluation of motivic functions, non-nullity, and integrability in fibers', Advances in Mathematics, Vol. 409, Part A, Paper No. 108635, 29 pages, doi:10.1016/j.aim.2022.108635 (2022)

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.

math.AG↗

arXiv2025

Serre's question on thin sets in projective space

We answer a question of Serre from the 1980s on rational points of bounded height on projective thin sets, in degree at least $4$. For degrees $2$ and $3$ we improve the known bounds in general. The focus is on thin sets of type II, namely corresponding to the images of ramified dominant quasi-finite covers of projective space, as thin sets of type I are already well understood via dimension growth results by the third author in 2002 (published in 2023) by a global variant of Heath-Brown's $p$-adic determinant method. For type II, we obtain a uniform affine variant of Serre's question which implies the projective case and for which the implicit constant is furthermore polynomial in the degree. We are able to avoid logarithmic factors when the degree is at least $5$ and we prove our results over any global field, of any characteristic. A key ingredient for obtaining the affine variant comes from Binyamini-Cluckers-Novikov (2024) and Binyamini-Cluckers-Kato (2025) where a question of the third author was answered by providing bounds, for rational points on irreducible curves, which are quadratic in the degree. A second key ingredient is an adaptation of Salberger's global determinant method to the case of weighted polynomials. The third key ingredient is the design of our affine variant of Serre's question, for weighted polynomials which are not necessarily weighted homogeneous.

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arXiv2024

Motivic Mellin transforms

This work brings Mellin transforms into the realm of motivic integration. The new, larger class of motivic functions is stable under motivic Mellin and Fourier transforms, with general Fubini results and change of variables formulas. It specializes to $p$-adic integrals and $p$-adic Mellin transforms uniformly in $p$, with transfer principles between zero and positive characteristic local fields. In particular, it generalizes previous set-ups of motivic integration with Fubini from among others [16, 17, 18, 29, 9] and simplifies some aspects on the way by using the ideas of [10].

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arXiv2024

Eventual tightness of projective dimension growth bounds: quadratic in the degree

In projective dimension growth results, one bounds the number of rational points of height at most $H$ on an irreducible hypersurface in $\mathbb P^n$ of degree $d>3$ by $C(n)d^2 H^{n-1}(\log H)^{M(n)}$, where the quadratic dependence in $d$ has been recently obtained by Binyamini, Cluckers and Kato in 2024 [1]. For these bounds, it was already shown by Castryck, Cluckers, Dittmann and Nguyen in 2020 [3] that one cannot do better than a linear dependence in $d$. In this paper we show that, for the mentioned projective dimension growth bounds, the quadratic dependence in $d$ is eventually tight when $n$ grows. More precisely the upper bounds cannot be better than $c(n)d^{2-2/n} H^{n-1}$ in general. Note that for affine dimension growth (for affine hypersurfaces of degree $d$, satisfying some extra conditions), the dependence on $d$ is also quadratic by [1], which is already known to be optimal by [3]. Our projective case thus complements the picture of tightness for dimension growth bounds for hypersurfaces.

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arXiv2024

Analytic holonomicity of real C$^{{\mathrm{exp}}}$-class distributions

We introduce a notion of distributions on $\mathbb{R}^n$, called distributions of C$^{{\mathrm{exp}}}$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C$^{{\mathrm{exp}}}$-class functions. We prove that the framework of C$^{{\mathrm{exp}}}$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C$^{{\mathrm{exp}}}$-class.

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arXiv2024

Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields

Let $C\subset{\mathbb P}_K^2$ be an algebraic curve over a number field $K$, and denote by $d_K$ the degree of $K$ over ${\mathbb Q}$. We prove that the number of $K$-rational points of height at most $H$ in $C$ is bounded by $c d^{2}H^{2d_K/d}(\log H)^\kappa$ where $c,\kappa$ are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties. The quadratic dependence on $d$ in the bound as well as the exponent of $H$ are optimal; the novel aspect is the quadratic dependence on $d$ which answers a question raised by Salberger. We derive new results on Heath-Brown's and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case $K={\mathbb Q}$. The proofs however are of a completely different nature, replacing the real analytic approach previously used by the $p$-adic determinant method. The optimal dependence on $d$ is achieved using a technical improvement in the treatment of high multiplicity points on mod $p$ reductions of algebraic curves.

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arXiv2023

Improvements on dimension growth results and effective Hilbert's irreducibility theorem

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by relaxing the condition on the top degree homogeneous part of the polynomial describing the affine hypersurface, while sharpening the dependence on the degree in the bounds compared to previous results. We formulate a conjecture about plane curves which provides a conjectural approach to the uniform degree $3$ case (the only remaining open case). For induction on dimension, we develop a higher dimensional effective version of Hilbert's irreducibility theorem, which is of independent interest.

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arXiv2023

Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on $d$, by showing the upper bound $C d^2 H^{2/d} (\log H)^κ$ with some absolute constants $C$ and $κ$. This bound is optimal with respect to both $d$ and $H$, except for the constants $C$ and $κ$. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the $H^ε$ factor by a power of $\log H$. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of Pólya, which allows us to save one extra power of $d$ compared with the standard approach using Bézout's theorem.

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arXiv2023

Hensel minimality II: Mixed characteristic and a diophantine application

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.

math.LO↗

arXiv2023

Parametric Fourier and Mellin transforms of power-constructible functions

We enrich the class of power-constructible functions, introduced in [CCRS23], to a class of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of the class. We furthermore identify a subclass which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This subclass is also stable under parametric integration, under taking pointwise and $L^p$limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in this subclass.

math.CA↗

arXiv2023

Existential uniform $p$-adic integration and descent for integrability and largest poles

Since the work by Denef, $p$-adic cell decomposition provides a well-established method to study $p$-adic and motivic integrals. In this paper, we present a variant of this method that keeps track of existential quantifiers. This enables us to deduce descent properties for $p$-adic integrals. In particular, we show that integrability for `existential' functions descends from any $p$-adic field to any $p$-adic subfield. As an application, we obtain that the largest pole of the Serre-Poincaré series can only increase when passing to field extensions. As a side result, we prove a relative quantifier elimination statement for Henselian valued fields of characteristic zero that preserves existential formulas.

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arXiv2023

Mellin transforms of power-constructible functions

We consider several systems of algebras of real- and complex-valued functions, which appear in o-minimal geometry and related geometrically tame contexts. For each such system, we prove its stability under parametric integration and we study the asymptotics of the functions as well as the nature of their parametric Mellin transforms.

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arXiv2022

Hensel minimality I

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.

math.LO↗

arXiv2022

The log canonical threshold and rational singularities

We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then ${\rm lct}(f,J_f^2)={\rm lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\widetilde{\alpha}(f)\geq{\rm lct}(f,J_f^2)$, where $\widetilde{\alpha}(f)$ is the minimal exponent of $f$. In the case of a polynomial over $\overline{\mathbf{Q}}$, we also prove an analogue of this latter inequality, with $\widetilde{\alpha}(f)$ replaced by the motivic oscillation index ${\rm moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-{\rm lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that ${\rm lct}(I)<{\rm lct}(J)$ for every $J$ with $I\subsetneq J$.

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arXiv2021

Evaluation of motivic functions, non-nullity, and integrability in fibers

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.

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arXiv2021

Point counting and Wilkie's conjecture for non-archimedean Pfaffian and Noetherian functions

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 Pila-Wilkie counting theorem in the archimean situation. Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${\mathbb{Q}}_p$, but the situation in ${\mathbb{C}}(\!(t)\!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a subanalytic set over ${\mathbb{C}}(\!(t)\!)$ is automatically finite, but give examples showing that their number may grow arbitrarily quickly even for analytic sets. Thus no analog of the Pila-Wilkie theorem can be expected to hold for general analytic sets. On the other hand we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in $r$, thus showing that the analog of Wilkie's conjecture does hold in this context.

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arXiv2021

A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points

We provide uniform estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka's resolution of singularities and Denef's formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called $E$-smooth morphisms ($E\in\mathbb{N}$), which refine the (FRS) property, and use the methods we developed to provide uniform number-theoretic estimates as above for their fibers. Similar estimates are given for fibers of $\varepsilon$-jet flat morphisms, improving previous results by the last two authors.

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arXiv2020

Combining Igusa's conjectures on exponential sums and monodromy with semi-continuity of the minimal exponent

We combine two of Igusa's conjectures with recent semi-continuity results by Musta\c{t}\u{a} and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in terms of degrees and dimensions only. We provide evidence consisting partly of adaptations of already known results about Igusa's conjecture on exponential sums, but also some new evidence like for all polynomials in up to $4$ variables. We show that, in turn, these bounds imply consequences for Igusa's (strong) monodromy conjecture. The bounds are related to estimates for major arcs appearing in the circle method for local-global principles.

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