No abstract

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for , where (resp.) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

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.

For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers pos of positive and neg of negative roots (counted with multiplicity) are majorized respectively by the numbers c and p of sign changes and sign preservartions in the sequence of its coefficients, and that the differences c − pos and p − neg are even numbers. For degree 5 polynomials, it has been proved by A. Albouy and Y. Fu that there exist no such polynomials having three distinct positive and no negative roots and whose signs of the coefficients are (+, +, −, +, −, −) (or having three distinct negative and no positive roots and whose signs of the coefficients are (+, −, −, −, −, +)). For degree 5 and when the leading coefficient is positive, these are all cases of numbers of positive and negative roots (all distinct) and signs of the coefficients which are compatible with Descartes' rule of signs, but for which there exist no such polynomials. We explain this non-existence and the existence in all other cases with d = 5 by means of pictures showing the discriminant set of the family of polynomials x 5 + x 4 + ax 3 + bx 2 + cx + d together with the coordinate axes.