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Floris Vermeulen

Publications and source records attributed to Floris Vermeulen.

At least 19 recordsLinked to original sources
arXiv2026

Ordered henselian valued fields: definability and Borel sets

We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning.

math.LO↗
arXiv2026

The Grothendieck ring of a non-divisible ordered abelian group is trivial

We consider the model-theoretic Grothendieck ring of definable sets in ordered abelian groups. It is well-known that $\mathrm{K} \mathbb{Q} \cong \mathbb{Z}[T]/(T^2 + T)$ and $\mathrm{K} \mathbb{Z} =0$, but surprisingly little is known about other cases. We present a short computation which shows that they all collapse: $\mathrm{K} G = 0$, unless $G$ is divisible.

math.LO↗
arXiv2026

Hensel minimality, $p$-adic exponentiation and Tate uniformization

We use Hensel minimality, a non-Archimedean analog of o-minimality, to study several questions around transcendental number theory, unlikely intersections, and differential fields in a non-Archimedean setting. In particular, we focus on $p$-adic exponentiation and Tate uniformization on $\mathbb{C}_p$, which we show live in a Hensel minimal structure on $\mathbb{C}_p$. We start by constructing a large collection of derivations on Hensel minimal fields that respect definable functions, which we then apply to the $p$-adic Schanuel conjecture. We also study properties of local definability in analogy to work of Wilkie, and show that $p$-adic Schanuel implies a uniform version of itself. For Tate uniformization we show a strong closure property when blurring, and deduce that $\mathbb{C}_p$ with the blurred Tate uniformization is quasiminimal. Finally, we prove a result on $p$-adic density of likely intersections for powers of elliptic curves.

math.LO↗
arXiv2025

Integration in Hensel minimal fields

We develop a framework of motivic integration in the style of Hrushovski--Kazhdan in arbitrary Hensel minimal fields of equicharacteristic zero. Hence our work generalizes that of Hrushovski--Kazhdan and Yin, but applies more broadly to discretely valued fields, almost real closed fields with analytic structure, pseudo-local fields, and coarsenings. In more detail, we obtain isomorphisms of Grothendieck rings of definable sets, with or without volume forms, in the valued field sort and in the leading term sort. Along the way we develop a theory of effective 1-h-minimal structures, where finite definable sets can be lifted from the leading term sort to the valued fields sort. We show that many natural examples of 1-h-minimal structures are effective, and develop dimension theory and a theory of differentiation in $\mathrm{RV}$ for effective structures.

math.LO↗
arXiv2025

Counting rational points on transcendental curves in valued fields

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.

math.NT↗
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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arXiv2025

A geometric determinant method and geometric dimension growth

We study a geometric version of the dimension growth conjecture. While it is closely related in spirit to themes arising in geometric Manin's conjecture, it applies in greater generality and provides more uniform bounds. For an irreducible projective variety $X$ defined over $\mathbb{C}(t)$, the set $X(b)$ of $\mathbb{C}(t)$-rational points on $X$ of degree less than $b$ has a natural structure of an algebraic variety over $\mathbb{C}$. We study the dimension and irreducibility of $X(b)$ when $X$ has degree $d \ge 2$, and obtain a geometric analogue of the classical dimension growth conjecture, namely that $\dim X(b) \le b\dim X $ for every $b \ge 1$. In particular, when $X$ is defined over $\mathbb{C}$, this provides uniform bounds on the dimension of the space of degree $b$ rational curves on $X$. We also develop a geometric version of Heath-Brown's $p$-adic determinant method for varieties defined over $\mathbb{C}(t)$. This allows us to show that as soon as $d \ge 6$, the number of irreducible components of $X(b)$ of dimension $b\dim X$ is bounded by a polynomial in $d$ which is independent of $b$. As a further application, we obtain an analogue of the Bombieri--Pila theorem for affine curves, as well as a corresponding result for projective curves.

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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].

math.AG↗
arXiv2024

Almost real closed fields with real analytic structure

Cluckers and Lipshitz have shown that real closed fields equipped with real analytic structure are o-minimal. This generalizes the well-known subanalytic structure $\mathbb{R}_{\mathrm{an}}$ on the real numbers. We extend this line of research by investigating ordered fields with real analytic structure that are not necessarily real closed. When considered in a language with a symbol for a convex valuation ring, these structures turn out to be tame as valued fields: we prove that they are $\omega$-h-minimal. Additionally, our approach gives a precise description of the induced structure on the residue field and the value group, and naturally leads to an Ax--Kochen--Ersov-theorem for fields with real analytic structure.

math.LO↗
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.

math.NT↗
arXiv2023

Dimension growth for affine varieties

We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in $\mathbb{A}^n$ which is not the preimage of a curve under a linear map $\mathbb{A}^n\to \mathbb{A}^{n-\dim X+1}$, then we prove that $X$ has at most $O_{d,n,\varepsilon}(B^{\dim X - 1 + \varepsilon})$ integral points up to height $B$. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning-Heath-Brown-Salberger. Our techniques follow the $p$-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck-Cluckers-Dittmann-Nguyen. The main difficulty is to count integral points on lines on an affine surface in $\mathbb{A}^3$, for which we develop point-counting results for curves in $\mathbb{P}^1\times \mathbb{P}^1$. We also formulate and prove analogous results over global fields, following work by Paredes-Sasyk.

math.NT↗
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

Scrollar invariants, syzygies and representations of the symmetric group II

Let $φ: C\to \mathbb{P}^1$ be a degree $d$ cover of curves. In work by Castryck, Zhao and the author, we showed how one can attach to each partition $λ$ of $d$ a multi-set of scrollar invariants of $λ$ with respect to $φ$. We studied these invariants when $φ$ is simply branched, and related these new scrollar invariants to known geometric data. In this article we show how one can remove this simple branching condition, using the notion of $S_d$-closure as developed by Bhargava and Satriano. With this new framework, we are able to generalize all results from this work to arbitrary covers $C\to \mathbb{P}^1$. In particular, we obtain a syzygy-free interpretation for the splitting types of the syzygy bundles in the Casnati--Ekedahl resolution of an arbitrary cover $φ: C\to \mathbb{P}^1$. By using the Maroni bound, we are able to give new general bounds on these splitting types.

math.AG↗
arXiv2023

Scrollar invariants, syzygies and representations of the symmetric group

We give an explicit minimal graded free resolution, in terms of representations of the symmetric group $S_d$, of a Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$ that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree $d$ cover of $\mathbb{P}^1$ by a relatively canonically embedded curve $C$, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree $4$ cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to $C \to \mathbb{P}^1$: one for each irreducible representation of $S_d$, i.e., one for each partition of $d$.

math.AG↗
arXiv2022

On abelian covers of the projective line with fixed gonality and many rational points

A smooth geometrically connected curve over the finite field $\mathbb{F}_q$ with gonality $γ$ has at most ${γ(q+1)}$ rational points. The first author and Grantham conjectured that there exist curves of every sufficiently large genus with gonality $γ$ that achieve this bound. In this paper, we show that this bound can be achieved for an infinite sequence of genera using abelian covers of the projective line. We also argue that abelian covers will not suffice to prove the full conjecture.

math.NT↗
arXiv2022

Hensel minimality: Geometric criteria for $\ell$-h-minimality

Recently, Cluckers, Halupczok and Rideau-Kikuchi developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this article, we continue the study of Hensel minimality, in particular focusing on $\omega$-h-minimality and $\ell$-h-minimality, for $\ell$ a positive integer. Our main results include an analytic criterion for $\ell$-h-minimality, preservation of $\ell$-h-minimality under coarsening of the valuation and $\ell$-dimensional dimensional geometry.

math.LO↗
arXiv2022

Curves of fixed gonality with many rational points

Given an integer $γ\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $γ$ and with exactly $γ(q+1)$ $\mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.

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