Diophantine properties of the zeros of (monic) polynomials the coefficients of which are the zeros of Hermite polynomials

TL;DR

This paper explores unique Diophantine properties of matrices constructed from polynomials whose coefficients are zeros of Hermite polynomials, revealing isospectral and integer-related eigenvalue characteristics.

Contribution

It introduces matrices derived from polynomials with Hermite zeros and proves their eigenvalues exhibit remarkable Diophantine and isospectral properties.

Findings

Eigenvalues of M_1 are the first N integers

Eigenvalues of M_2 are the first N squared-integers

Technique extends to other polynomial families

Abstract

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.