On Hurwitz Stable Polynomials with Integer Coefficients

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.