Roots of Polynomials with Positive Coefficients Erik I. Verriest 1 and Nak-seung Patrick Hyun 2 Abstract— A necessary condition for stability of a finite- dimensional linear time-invariant system is that all the co- efficients of the characteristic equation are strictly positive. However, it is well-known that this condition is not sufficient, except for n less than 3. In this paper, we show that any polynomial that has positive coefficients cannot have roots on the nonnegative real axis. Conversely, if a polynomial has no roots on the positive real axis, a polynomial with positive coefficients can be found so that the product of the two polynomials also has positive coefficients. A simple upper bound for the degree of this multiplier polynomial is given. One application of the main result is that under a strict condition, it is possible to find a non-minimal realization of a given transfer function using only positive multipliers (except for the “minus” in the standard feedback comparator). Keywords: positive polynomial, root locations, positive-real, positive system AMS Classification: 12D10, 26C10, 93D99 I. INTRODUCTION All coefficients of the polynomial a(s)= s 6 +4s 5 +3s 4 +2s 3 + s 2 +4s +4 (Example 3.30 in [8]) are positive. Positivity of the coefficients is a necessary condition to have all its roots in the open left half plane. This condition is not sufficient, as is the case in the above polynomial. By performing the Routh-Hurwitz test, it can be seen that this polynomial will have two roots in the right-half plane. This prompts the question as to where precisely the roots of a polynomial can or cannot lie. This short note resolves this question by identifying such a root property. It also goes beyond, and shows that any polynomial satisfying this root property is a factor of a polynomial with all positive coefficients. Some results related to this problem appeared in [2]. The authors asked the question if a conjugate pair of zeros can be factored out from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonneg- ative coefficients. Sendov [14] applies their work to prove a Gauss-Lucas type theorem for non convex (symmetric w.r.t. R) sectors in C. Much earlier, Obrechkoff proved an upper bound on the number of roots in the sector {s ∈ C \{0}| arg(s)| <θ} for θ ∈ (0,π/2). The bound implies *This work was supported by NSF grant: CPS 1544857 1 Erik I. Verriest is with the Faculty of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 erik.verriest@ece.gatech.edu 2 Nak-seung Patrick Hyun is a PhD student at Georgia Institute of Technology, Atlanta, GA 30332-0250 pathyun@gmail.com that no roots can lie on R + . Handelman [9] showed that for a monic (Laurent) polynomial, p ∈ R[s], with non-negative roots, there exists a positive integer m such that (1+s) m p(s) has only positive coefficients. A short proof of this fact is given by Akiyama [1]. In this paper we give an easily computable bound for m and show its asymptotic behavior as the roots of p get closer to the real axis. Dubickas [7] solves an algebraic number theoretic problem, for which he proved the auxiliary lemma: Lemma [7]: Let s 0 ∈ C \ R. Then there exists a monic polynomial q ∈ Q[s] such that the polynomial (s − s 0 )(s − s 0 )q(s) has positive coefficients. This settled a conjecture by Kuba [11]. This problem is further studied by Borel [3], Za¨ ımi [15], and Brunotte. In [4], Brunotte finds for p of second degree, the lowest degree, δ, of q such that pq has positive coefficients and the lowest degree, δ 0 , of a polynomial q 0 such that pq 0 has nonnegative coefficients, thus extending a result of Meissner [12], who constrained q to have positive coefficients. Brunotte shows that these polynomials can be calculated in finitely many steps. The exact formula for δ when deg(p) > 2 is still unknown. In [5] it is shown that if p ∈ Z[s], a polynomial q ∈ Z[s] exists. Our contribution is to determine a readily computable polynomial although not of minimal degree that will solve the problem, make connections with positive realness, and look at possible applications to positive system theory. A. Notation for some important point sets If x ∈ C, then its real and imaginary parts are denoted by respectively ℜ(x) and ℑ(x). R − = {x ∈ R|x< 0} R + = {x ∈ R|x> 0} C − = {x ∈ C|ℜ(x) < 0} C+ = {x ∈ C|ℜ(x) > 0} C 0 = {x ∈ C|ℜ(x)=0} C s− = {x ∈ C|ℜ(x) < 0}\ R − C s+ = {x ∈ C|ℜ(x) > 0}\ R + C s0 = {x ∈ C|ℜ(x)=0}\{0} The closure of these sets (whenever appropriate) will be denoted by an overbar. With these number sets, the complex plane C partitions as C = {0}∪ R − ∪ R + ∪ C s− ∪ C s+ ∪ C s0 . 23rd International Symposium on Mathematical Theory of Networks and Systems Hong Kong University of Science and Technology, Hong Kong, July 16-20, 2018 259