A converse rule of signs for polynomials (Q1058001)

Given a polynomial P(x), it is well-known that the Cardano-Descartes rule of signs gives only an upper bound on the number of its positive roots, except in the case in which there is one or no sign variation, where it indicates that P(x) has one or no positive root(s) respectively. In certain new root isolation methods, of great interest to symbolic mathematical computation, it is important to know under what conditions the existence of one positive root implies that P(x) presents only one sign variation. These conditions are discussed and presented in this paper.