(untitled) (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2019)
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 posc( mod 2) and negp( 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.