Polynomials, sign patterns and Descartes' rule of signs

Abstract

By Descartes' rule of signs, a real degree d polynomial P with all nonvanishing coefficients, with c sign changes and p sign preservations in the sequence of its coefficients (c + p = d) has pos ≤ c positive and neg ≤ p negative roots, where pos ≡ c( mod 2) and neg ≡ p( mod 2). For 1 ≤ d ≤ 3, for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair (pos, neg) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly neg negative roots (all of them simple). For d ≥ 4 this is not so. It was observed that for 4 ≤ d ≤ 10, in all nonrealizable cases either pos = 0 or neg = 0. It was conjectured that this is the case for any d ≥ 4. We show a counterexample to this conjecture for d = 11. Namely, we prove that for the sign pattern (+, −, −, −, −, −, +, +, +, +, +, −) and the pair (1, 8) there exists no polynomial with 1 positive, 8 negative simple roots and a complex conjugate pair.

Descartes' rule of signs and moduli of roots

“…(3) For real, but not necessarily hyperbolic degree d polynomials, one can ask the question: It seems that the question has been explicitly formulated for the first time in [2]. The answer to it is not trivial and the exhaustive one is known for d ≤ 8, see [7], [1], [5], [8] and [9]. The proof of the realizability of certain cases is often done by means of a concatenation lemma, see Lemma 2 in Section 7.…”

New Aspects of Descartes’ Rule of Signs

“…The case d = 8 has been partially solved in [8] and completely in [16]: Observe that for d ≤ 8, all examples of couples (SP, AP) which are non-realizable are with APs of the form (ν, 0) or (0, ν), ν ∈ N. Initially we thought that this is always the case. However recently it was proven that, for higher degrees, this fact is no longer true, see [17]: There is a strong evidence that for d = 9, the similar couple (SP, AP) ((+, −, −, −, −, +, +, +, +, −) , (1, 6)) is also non-realizable. (Its Descartes' pair equals (3,6).)…”

“…If this were true, then 9 would be the smallest degree with an example of a non-realizable couple (SP, AP) for which both components of the AP are nonzero. When studying the cases d = 8 and d = 11 (see [16] and [17]) discriminant sets have been considered, see Remark 2.…”

A non-realization theorem in the context of Descartes' rule of signs

“…The following lemma on hyperbolic polynomials is proved in [11]. It is used in the proofs of the other lemmas.…”

“…Remark 3. It is shown in [11] that for d = 11, the admissible pair (1,8) is not realizable with the sign pattern (+ -----+ + + + + -). Hence Theorem 1 shows an example of a nonrealisable couple, with both components of the admissible pair different from zero, in the least possible degree (namely, 9).…”