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

Abstract

For 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), Descartes' rule of signs says that P 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 neg negative roots (all of them simple); that is, all these cases are realizable. This is not true for d ≥ 4, yet for 4 ≤ d ≤ 8 (for these degrees the exhaustive answer to the question of realizability is known) in all nonrealizable cases either pos = 0 or neg = 0. It was conjectured that this is the case for any d ≥ 4. For d = 9, we show a counterexample to this conjecture: for the sign pattern (+, −, −, −, −, +, +, +, +, −) and the pair (1, 6) there exists no polynomial with 1 positive, 6 negative simple roots and a complex conjugate pair and, up to equivalence, this is the only case for d = 9.

Real univariate polynomials with given signs of coefficients and simple real roots

“…At present the exhaustive answer is known up to degree 8 as well as several infinite series of non-realizable triples (σ, (ℓ + , ℓ − )), see e.g. [3] and the references therein.…”