Budan tables of real univariate polynomials

The Budan table of f collects the signs of the iterated derivatives of f. We revisit the classical Budan–Fourier theorem for a univariate real polynomial f and establish a new connectivity property of its Budan table. We use this property to characterize the virtual roots of f (introduced by Gonzalez-Vega, Lombardi, Mahé in 1998); they are continuous functions of the coefficients of f. We also consider a property (P) of a polynomial f, which is generically satisfied, it eases the topological-combinatorial description and study of 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 finite stratification of the space of polynomials with fixed degree.