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<p>A <b>Littlewood polynomial</b> is a univariate polynomial all of whose coefficients lie in {±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.</p>

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Counting (skew-)reciprocal Littlewood polynomials with square discriminant

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  • 发表日期：

    2026年4月6日

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  • David Hokken

摘要

A Littlewood polynomial is a univariate polynomial all of whose coefficients lie in {±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.