On Hurwitz Stable Polynomials with Integer Coefficients

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.