Budan Tables of Real Univariate Polynomials

International audienceThe Budan table of f collects the signs of the iterated derivative of f. We revisit the classical Budan-Fourier theorem for a univariate real polynomial f and establish a new connexity property of its Budan table. We use this property to characterize the virtual roots of f, (introduced by Gonzales-Vega, Lombardi, Mahe in 1998); they are continuous functions of the coecients of f. We also consider a property (P) of a polynomial f, which is generically satis ed, it eases the topological-combinatorial description and study of the Budan tables. A natural extension of the information collected by the virtual roots provides alternative representations of (P)-polynomials; while an attached tree structure allows a strati fication of the space of (P)-polynomials. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.La table de Budan collecte les signes des derivees du polynome f, c'est le tableau de variation de f. Son etude permet de definir de nouveaux inavariants et une stratification de l'espace des polynomes