Global bounds on stable polynomials

TL;DR

This paper improves classical bounds on stable polynomials, extends Szász inequalities to multiple variables using determinantal representations, and demonstrates how two-variable inequalities can inform bounds in higher dimensions.

Contribution

It advances the theory of polynomial stability by refining inequalities and establishing new links between single and multivariable cases using determinantal methods.

Findings

Improved Szász bounds for stable polynomials.

Derived Szász-type inequalities in two variables.

Extended inequalities to several variables using two-variable results.

Abstract

A classical inequality of Sz\'asz bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea-Br\"and\'en generalized this result to several variables as a piece of their characterization of linear maps on polynomials preserving stability. In this paper, we improve Sz\'asz's original inequality, use determinantal representations to prove Sz\'asz type inequalities in two variables, and then prove that one can use the two variable inequality to prove an inequality for several variables.