Counting (skew-)reciprocal Littlewood polynomials with square discriminant

TL;DR

This paper derives the asymptotic count of reciprocal and skew-reciprocal Littlewood polynomials with square discriminant, linking to Galois group conjectures and Pythagorean triples.

Contribution

It provides the first asymptotic formulas for these specialized polynomials with square discriminant, connecting polynomial enumeration with number theory conjectures.

Findings

Asymptotic formulas for reciprocal Littlewood polynomials with square discriminant

Connection to a bounded-height Van der Waerden conjecture on Galois groups

Asymptotics of Gaussian-weighted counts of Pythagorean triples

Abstract

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.