Counting (skew-)reciprocal Littlewood polynomials with square discriminant

2023/01/13 by Hokken, David · 3 citations
#05A16. Secondary: 11P21 #11R09 #11R32 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11C08

paper · doi:10.48550/arxiv.2301.05656

Abstract

A Littlewood polynomial is a single-variable 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.

Cited by

• Irreducibility and Galois groups of random reciprocal polynomials of large degree
• Law of large numbers for the discriminant of random polynomials
• Galois groups of random integer matrices

Related

• Do flat skew-reciprocal Littlewood polynomials exist?
• Asymptotic L⁴ norm of polynomials derived from characters
• Littlewood Polynomials with Small L⁴ Norm
• Algebraic aspects of the polynomial Littlewood-Offord problem
• Irreducibility of Littlewood polynomials of special degrees