Circulants and critical points of polynomials

TL;DR

This paper establishes a connection between circulant matrices and polynomial critical points, providing new proofs and generalizations of Schoenberg-type conjectures, and exploring majorization properties of polynomial critical points.

Contribution

It proves that the spectrum of a submatrix of a circulant matrix equals the critical points of its characteristic polynomial, and generalizes Schoenberg-type conjectures for polynomials with zero mass center.

Findings

Spectrum of submatrix equals polynomial critical points.

Provided a simple proof of Schoenberg conjecture.

Generalized Schoenberg-type inequalities for zero mass center polynomials.

Abstract

We prove that for any circulant matrix $C$ of size $n\times n$ with the monic characteristic polynomial $p(z)$, the spectrum of its $(n-1)\times(n-1)$ submatrix $C_{n-1}$ constructed with first $n-1$ rows and columns of $C$ consists of all critical points of $p(z)$. Using this fact we provide a simple proof for the Schoenberg conjecture recently proved by R. Pereira and S. Malamud. We also prove full generalization of a higher order Schoenberg-type conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S. Cheung and T.W. Ng. in its original form, i.e. for polynomials whose mass centre of roots equals zero. In this particular case, our inequality is stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser's-like results on majorization of critical point of polynomials are also obtained.