More on the Hankel Pencil Conjecture—News on the Root Conjecture

Abstract

The Hankel pencil conjecture concerns certain pencils of $n\times n$ Hankel matrices and has a control theoretic origin; see [4]. For each specific n it was abbreviated in [2] as HPnC and reduced to a conjecture RnC about roots of pairs of certain polynomials of degree $n-2$. To be solved, each conjecture RnC would be laboriously translated into a system of equations for the elementary symmetric polynomials and solved by Gröbner basis methods (we stopped at $n=8$). In this paper we present conjecturally a parametrized system of equations in the symmetric polynomials which permits to prove specific cases of the root conjecture and hence of the Hankel pencil conjecture by much lighter computation. Other formulations of the root conjecture are also given.