Searcharxiv⌕ Search

arXiv subjects

Dante Bonolis

Publications and source records attributed to Dante Bonolis.

11 recordsLinked to original sources
arXiv2026

Genuine and strongly genuine polynomials: With an application to the persistence of Galois groups under specialization

We develop the theory of strongly $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}'=(x_2,\ldots,x_n) \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ is reducible over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) can be well-controlled quantitatively. We also develop the theory of a larger class of $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}' \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ splits completely over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) into factors that are linear in $Y$ can be well-controlled quantitatively. For each of these classes, we prove that there are four equivalent characterizations. As an application, we demonstrate that $n$-genuine and strongly $n$-genuine polynomials can be used to prove, for any polynomial $F(Y,X_1,\ldots,X_n)$, an upper bound for the number of specializations $F(Y,\mathbf{x})$ with $\mathbf{x}=(x_1,\ldots,x_n) \in \mathbb{Z}^n$ such that the Galois group of the splitting field of $F(Y,\mathbf{x})$ over $\mathbb{Q}$ is not isomorphic to the Galois group of the splitting field of $F(Y,X_1,\ldots,X_n)$ over $\mathbb{Q}(X_1,\ldots,X_n)$. We simultaneously prove analogous results over any number field.

math.NT↗
arXiv2026

Counting points in thin sets: A survey

In the 1980's Serre asked how many points of bounded height can lie in a thin set. This has motivated significant research ever since, culminating in a series of recent breakthroughs. It is a good time to take stock of the central questions that have been resolved, and also to highlight remaining open questions. First, we survey recent progress on counting points of bounded height in the four types of thin sets, according to the projective/affine and type I/type II designations. Second, we turn to questions of uniformity. Famously, in the setting of type I thin sets, the best-known upper bound for the number of points of bounded height is independent of the maximum size, say $\|F\|$, of the coefficients of the polynomials that define the thin set; such an upper bound is called uniform. A uniform upper bound in the setting of type II thin sets is not known. For type II thin sets, we explore the dependence on $\|F\|$ via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.

math.NT↗
arXiv2025

On the $2$-torsion in class groups of number fields

In $2020$, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension $K/\mathbb{Q}$ of degree $n\geq 5$, the size of the $2$-torsion class group is bounded by $\# h_{2}(K)=O_{n,\varepsilon}(D_{K}^{\frac{1}{2}-\frac{1}{2n}+\varepsilon})$, where $D_{K}$ is the absolute discriminant of $K$. In the present paper, we improve their bound by proving that $\# h_{2}(K)=O_{n,\varepsilon}(D_{K}^{\frac{1}{2}-\frac{1}{2n}-\delta_{K}+\varepsilon})$, for a constant $\delta_{K}\geq\frac{1}{28n}-\frac{3}{28n(n-1)}$.

math.NT↗
arXiv2025

Stratification theorems for exponential sums in families

We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis, Pierce and Woo (arXiv:2505.11226), we prove that these stratification statements admit uniform variants in families, both algebraically and analytically. The paper includes an Appendix by Forey, Fres\'an and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.

math.NT↗
arXiv2025

Counting integral points in thin sets of type II: singularities, sieves, and stratification

Consider an absolutely irreducible polynomial $F(Y,X_1,\ldots,X_n) \in \mathbb{Z}[Y,X_1,\ldots,X_n]$ that is monic in $Y$ and is a polynomial in $Y^m$ for an integer $m \geq 1$. Let $N(F,B)$ count the number of $\mathbf{x} \in [-B,B]^n \cap \mathbb{Z}^n$ such that $F(y,\mathbf{x})=0$ is solvable for $y \in\mathbb{Z}$. In nomenclature of Serre, bounding $N(F,B)$ corresponds to counting integral points in an affine thin set of type II. Previously, in this generality Serre proved $N(F,B) \ll_F B^{n-1/2}(\log B)^{\gamma}$ for some $\gamma<1$. When $m \geq 2$, this new work proves $N(F,B) \ll_{n,F,\epsilon} B^{n-1+1/(n+1) + \epsilon}$ under a nondegeneracy condition that encapsulates that $F(Y,\mathbf{X})$ is truly a polynomial in $n+1$ variables, even after performing any $\text{GL}_n(\mathbb{Q})$ change of variables on $X_1,\ldots,X_n$. Under GRH, this result also holds when $m=1$. We show that generic polynomials satisfy the relevant nondegeneracy condition. Moreover, for a certain class of polynomials, we prove the stronger bound $N(F,B) \ll_{F} B^{n-1}(\log B)^{e(n)}$, comparable to a conjecture of Serre. A key strength of these results is that they require no nonsingularity property of $F(Y,\mathbf{X})$. The Katz-Laumon stratification for character sums, in a new uniform formulation appearing in a companion paper of Bonolis, Kowalski and Woo, is a key ingredient in the sieve method we develop to prove upper bounds that explicitly control any dependence on the size of the coefficients of $F$.

math.NT↗
arXiv2022

Application of a polynomial sieve: beyond separation of variables

Let a polynomial $f \in \mathbb{Z}[X_1,\ldots,X_n]$ be given. The square sieve can provide an upper bound for the number of integral $\mathbf{x} \in [-B,B]^n$ such that $f(\mathbf{x})$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=y^r$ is solvable for $y \in \mathbb{Z}$; then to a polynomial sieve, counting $\mathbf{x} \in [-B,B]^n$ for which $f(\mathbf{x})=g(y)$ is solvable, for a given polynomial $g$. Formally, a polynomial sieve lemma can encompass the more general problem of counting $\mathbf{x} \in [-B,B]^n$ for which $F(y,\mathbf{x})=0$ is solvable, for a given polynomial $F$. Previous applications, however, have only succeeded in the case that $F(y,\mathbf{x})$ exhibits separation of variables, that is, $F(y,\mathbf{x})$ takes the form $f(\mathbf{x}) - g(y)$. In the present work, we present the first application of a polynomial sieve to count $\mathbf{x} \in [-B,B]^n$ such that $F(y,\mathbf{x})=0$ is solvable, in a case for which $F$ does not exhibit separation of variables. Consequently, we obtain a new result toward a question of Serre, pertaining to counting points in thin sets.

math.NT↗
arXiv2022

Density of rational points on some quadric bundle threefolds

We prove the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1,2). The proof uses a mixture of the circle method and techniques from the geometry of numbers.

math.NT↗
arXiv2021

The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$-periodic complex valued functions which satisfy our conditions, but for which the Pólya-Vinogradov inequality is sharp.

math.NT↗
arXiv2019

A Polynomial Sieve and Sums of Deligne Type

Let $f\in\mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in\mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

math.NT↗
arXiv2018

On the size of the maximum of incomplete Kloosterman sums

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq H 0$ there exists some $a\in\mathbb{F}_{p}^{\times}$ such that \[ M(e(\tfrac{ax+\overline{x}}{p}))\geq \Big(\frac{1-\varepsilon}{\sqrt{2}π}+o(1)\Big)\log\log p. \] Moreover we also provide some result on the growth of the moments of $\{M(e(\tfrac{ax+\overline{x}}{p}))\}_{a\in\mathbb{F}_{p}^{\times}}$.

math.NT↗