A Root Isolation Algorithm for Sparse Univariate Polynomials

We consider a univariate polynomial f with real coefficients having a high degree $N$ but a rather small number $d+1$ of monomials, with $d\ll N$. Such a sparse polynomial has a number of real root smaller or equal to $d$. Our target is to find for each real root of $f$ an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the sparse structure. Our approach relies on the generalized Budan-Fourier theorem of Coste, Lajous, Lombardi, Roy and the techniques developed in some previous works of Galligo. To such a $f$ is associated a set of $d + 1$ $\mathbb{F}$-derivatives. The Budan-Fourier function $V_f(x)$ counts the sign changes in the sequence of $\mathbb{F}$-derivatives of the $f$ evaluated at $x$. The values at which this function jumps are called the $\mathbb{F}$-virtual roots of $f$, these include the real roots of $f$. We also consider the augmented $\mathbb{F}$-virtual roots of $f$ and introduce a genericity property which eases our study. We present a real root isolation method and an algorithm which has been implemented in Maple. We rely on an improved generalized Budan-Fourier count applied to both the input polynomial and its reciprocal, together with Newton like approximation steps.