On solving univariate sparse polynomials in logarithmic time

Authors:
J. Maurice Rojas;Yinyu Ye
Affiliations:
Department of Mathematics, Texas A&M University, TAMU 3368, College Station, TX;Department of Management Science and Engineering, School of Engineering, Stanford University, Stanford, CA
Venue:
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Year:
2005

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Abstract

Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m = 3 we can approximate within ε all the roots of f in the interval [0,R] using just O(log(D)log(D log R/ε)) arithmetic operations. In particular, we can count the number of roots in any bounded interval using just O(log2 D) 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.