New Designs for the Descartes Rule of Signs

Abstract

Descartes's rule of signs says that the number of positive roots of p(x) is equal to the number of sign changes in the sequence ao, al, ... , an, or is less than this number by a positive even integer. Investigating which of the possible numbers of roots permitted by Descartes actually occur, Anderson, Jackson, and Sitharam [1] show that the rule cannot be improved. For any sign sequence not containing zero, they construct polynomials with this sign sequence in the coefficients and any prescribed number of positive roots that is in accord with Descartes's rule. Analogously, since the negative roots are the roots of p(-x), all allowable numbers of negative roots are realized. Grabiner [2] establishes further examples. In particular, he provides polynomials for sign sequences that may contain zero and shows how to achieve certain numbers of positive and negative roots simultaneously.