Improved Budan-Fourier Count for Root Finding

Given a degree n univariate polynomial f(x), the Budan-Fourier function Vf (x) counts the sign changes in the sequence of derivatives of f evaluated at x. The values at which this function jumps are called the virtual roots of f , these include the real roots of f and any multiple root of its derivatives. This concept was introduced (by an equivalent property) by Gonzales-Vega, Lombardi, Mahé in [17], and then studied by Coste, Lajous, Lombardi, Roy in [8]. The set of virtual roots provide a good real substitute to the set of complex roots; it depends continuously on the coefficients of f . We will describe a root isolation method by a subdivision process based on a generalized Budan-Fourier count, fast evaluation and Newton like approximations. Our algorithm will provide isolating intervals for all augmented virtual roots of f . For a polynomials with integer coefficients of length size τ = Õ(n), its bit cost is in Õ(n). We rely on a new connexity property of the Budan table of f which collects the signs of the iterated derivatives of f . keywords: real univariate polynomial; real root isolation; refinement; BudanFourier theorem; Descartes rule; virtual roots; Budan table; Newton process; multiple roots; discretization; separation bound.