arXiv subjects

Tijs Buggenhout

Publications and source records attributed to Tijs Buggenhout.

3 recordsLinked to original sources

arXiv2026

Approximating parametric suprema for constructible and power-constructible functions

We prove that one may approximate parametric suprema of constructible and power-constructible functions using functions within the same class. This resolves a conjecture by Adiceam and Cluckers, which was posited after studying a question posed by Sarnak. We apply our result to prove that a certain subclass of Cexp-class distributions is tempered and to make uniform a bound concerning pushforward measures.

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.

math.NT↗

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.

math.NT↗