The irreducibility of some Wronskian Hermite polynomials

TL;DR

This paper investigates the irreducibility of certain Wronskian Hermite polynomials associated with partitions, establishing conditions under which they are irreducible and exploring implications for their zeros.

Contribution

It introduces new theorems providing bounds and congruences for Wronskian Hermite polynomials, advancing understanding of their algebraic properties and zero distributions.

Findings

Remainder polynomial is irreducible for partitions (n,m) with m <= 2

Remainder polynomial is irreducible for (n,n) when n+1 is a perfect square

Proves Veselov's conjecture for specific partitions

Abstract

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the partitions (n, m) with m <= 2, and (n, n) when n + 1 is a square. Our main tools are two theorems that we prove for all partitions. The first result gives a sharp upper bound for the slope of the edges of the Newton polygon for the remainder polynomial. The second result is a Schur-type congruence for Wronskian Hermite polynomials. We also explain how irreducibility determines the number of real zeros of Wronskian Hermite polynomials, and prove Veselov's conjecture for partitions of the form (n, k, k-1, ..., 1).