New Designs for the Descartes Rule of Signs

The question of how to construct polynomials having as many roots as allowed by the Descartes rule of signs has been the focus of interest recently 1],,2]. For a given real polynomial p(x) = a 0 + a 1 x + + a n x n ; 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 a 0 ; a 1 ; : : : ; a n , 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 coeecients 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. In this note we add to the previous work by presenting more general designs for given numbers of roots that yield a richer class of functions. Their construction might be more intuitive than earlier methods and, once one has the technique at hand, easier to do. Further, they give rise to a counterexample to a conjecture by Grabiner 2] concerning the numbers of positive and negative roots. The basic idea follows Grabiner, who uses the fact that a continuous function must have a root between points of opposite sign. Instead of constructing polynomials directly, however, we rst look at a diierent class of functions. We