arXiv subjects

David Hokken

Publications and source records attributed to David Hokken.

7 recordsLinked to original sources

arXiv2026

The strong form of Van der Waerden's conjecture via twisted Chowla

Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \geq 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_j| \leq B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B) \ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B) = o(B^{n-1})$. Inspired by recent progress, we now address this strong form. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. Assuming a twisted function field version of Chowla's conjecture, we prove this claim. This not only connects two active and challenging areas of research, but also conditionally resolves the strong Van der Waerden conjecture for all $n \geq 7$. Our proof is based on a variant of Heath-Brown and Pierce's square sieve and $q$-van der Corput differencing. Our methods also apply to the analogous problem of counting square discriminants of polynomials that are not necessarily monic. In addition to describing our new contributions, we briefly elaborate on the various conjectures appearing in Van der Waerden's paper and some of the exciting recent work of others in this area of arithmetic statistics.

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arXiv2026

Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(\chi_{-f},-1)$. The strong form of the conjecture was previously known to hold for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.

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arXiv2025

Irreducibility and Galois groups of random reciprocal polynomials of large degree

Let $A = a_0T^m + \sum_{j=1}^{m-1} a_j (T^{m-j}+T^{m+j}) + T^{2m}+1 \in \mathbf{Z}[T]$ be a monic reciprocal polynomial of degree $2m$ sampled randomly by selecting its coefficients $a_0,a_1,\dots,a_{m-1}$ independently according to a given probability measure $\mu$ on $\mathbf{Z}$. For a wide range of measures $\mu$, we prove that $A$ is irreducible with probability $\ge 1-Cm^{-c}$ for some absolute constants $c,C>0$. In addition, we prove that with the same probability the Galois group of $A$ is either the full hyperoctahedral group $\mathcal{C}_2 \wr \mathcal{S}_m$ or one of two of its index-$2$ subgroups. The main condition that $\mu$ must satisfy is of Fourier-theoretic nature, and holds for example when $\mu$ is the uniform measure on a set of at least $35$ consecutive integers, or on an arbitrary, sufficiently large subset of an interval $[-H,H]$, with $H$ larger than some absolute constant. Our most general result allows for each $a_j$ to be sampled by its own probability measure $\mu_j$. Our approach builds on earlier work of Bary-Soroker, Kozma and the second author, who proved for essentially the same $\mu_j$ that the 'standard' monic polynomial $a_0 + \cdots + a_{m-1}T^{m-1} + T^m$ is irreducible and has as Galois group either the symmetric group $\mathcal{S}_m$ or the alternating group $\mathcal{A}_m$ with high probability, conditioning on $a_0 \neq 0$. In our setting of reciprocal polynomials, we can rule out (all subgroups of) the maximal alternating subgroup $(\mathcal{C}_2 \wr \mathcal{S}_m) \cap \mathcal{A}_{2m}$ of the hyperoctahedral group as likely Galois group of $A$ by analyzing its discriminant.

math.NT↗

arXiv2025

The asymptotic Mahler measure of Gaussian periods

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the (multivariate) Mahler measure of a family of log Calabi-Yau varieties of increasing dimension. In turn, we study the asymptotics of some of these Mahler measures as the dimension increases, as well as properties of the associated algebraic dynamical system. We describe computational experiments that suggest that these cyclotomic integers realise the smallest non-zero logarithmic Mahler measure in the set of algebraic integers with cyclic Galois group of a given odd order. Finally, we discuss some precise conjectures that imply double logarithmic growth for those Mahler measures as a function of that order. The proofs use ideas from the theory of quantitative equidistribution, reflexive polytopes and toric varieties, the theory of random walks, Bessel functions, class field theory, and Linnik's constant.

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arXiv2024

Topology of zero sets of polynomials with square discriminant

Let $\mathcal{N} \neq \{0\}$ be a fixed set of integers, closed under multiplication, closed under negation, or containing $\{\pm 1\}$. We prove that any zero of a polynomial in $\mathbf{Z}[X]$ whose coefficients lie in $\mathcal{N}$ can be approximated in $\mathbf{C}$ to arbitrary precision by a zero of a polynomial in $\mathbf{Z}[X]$ with square discriminant whose coefficients also lie in $\mathcal{N}$. Hence the topology of the closure in $\mathbf{C}$ of the set of zeros of all such polynomials is insensitive to the discriminant being a square, in contrast to the Galois groups of the polynomials.

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arXiv2023

Irreducibility of Littlewood polynomials of special degrees

Let $f$ be sampled uniformly at random from the set of degree $n$ polynomials whose coefficients lie in $\{ \pm 1\}$. A folklore conjecture, known to hold under GRH, states that the probability that $f$ is irreducible tends to $1$ as $n$ goes to infinity. We prove unconditionally that $$\limsup_{n \to \infty} \mathbb{P}(f \text{ is irreducible}) = 1.$$

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arXiv2023

Counting (skew-)reciprocal Littlewood polynomials with square discriminant

A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant. This relates to a bounded-height analogue of the Van der Waerden conjecture on Galois groups of random polynomials. As a byproduct, we establish the asymptotics of certain Gaussian-weighted counts of Pythagorean triples.

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