On Hurwitz stable polynomials with integer coefficients

TL;DR

This paper investigates Hurwitz stable polynomials with positive integer coefficients, focusing on minimizing the largest coefficient and the sum of coefficients, establishing the existence of limits for these minimal values and deriving bounds.

Contribution

It introduces the concepts of h(N) and s(N) for minimal coefficients and sums, proves the existence of their limits using Fekete's lemma, and provides tight bounds for these limits.

Findings

Limits of the N-th roots of h(N) and s(N) exist and coincide.

The paper derives tight bounds for the common limit value.

It characterizes minimal coefficient polynomials with Hurwitz stability.

Abstract

Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for polynomials in H(N) with minimal sum of the coefficients. Let h(N) and s(N) denote these minimal values. Using Fekete's subadditive lemma we show that the N-th square roots of h(N) and s(N) have a limit as N goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.