On solving univariate sparse polynomials in logarithmic time

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( log 2 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.