On arrangements of roots for a real hyperbolic polynomial and its derivatives

Abstract

In this paper we count the number ♯ n (0, k) , k⩽ n−1, of connected components in the space Δ n (0, k) of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p( x) together with all its nonvanishing derivatives under the assumption that the roots of p( x) are real. Already the first nontrivial case n=4 shows that the obvious restrictions coming from the standard Rolle's theorem are insufficient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials.