Which sign patterns are canonical? (Q2093072)

The paper deals with real polynomials \(P(x)=a_d x^d+\dots +a_1 x+a_0\) without vanishing coefficients, whose all roots are real. Assume that the moduli of all roots are distinct. In one of his previous works, the author showed that the signs of the coefficients generate some canonical order of the moduli of the roots on the real positive half-line. The sign pattern of coefficients is called canonical if it is realizable only by its canonical order of the moduli of roots. In the present paper, the author gives the exhaustive description of all canonical patterns of coefficients: A sign pattern is canonical if and only if it contains none of the configurations \((++--), (--++), (+--+), (-++-).\)