On solving univariate sparse polynomials in logarithmic time

Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m 1⁄4 3 we can approximate within e all the roots of f in the interval 1⁄20;R using just OðlogðDÞlogðD log Re ÞÞ arithmetic operations. In particular, we can count the number of roots in any bounded interval using just Oðlog DÞ arithmetic operations. Our speed-ups are significant and near-optimal: The asymptotically sharpest previous complexity upper bounds for both problems were super-linear in D; while our algorithm has complexity close to the respective complexity lower bounds. We also discuss conditions under which our algorithms can be extended to general m; and a connection to a real analogue of Smale’s 17th Problem. r 2004 Elsevier Inc. All rights reserved.