ON SOLVING FEWNOMIALS OVER INTERVALS IN FEWNOMIAL TIME

Let f be a degree D univariate polynomial with real coefficients and exactlym monomial terms. We show that in the special case m=3 we can approximate within e all the roots of f in the interval (0, R) using just O log(D)log Dlog R e �� 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 dis- cuss conditions under which our algorithms can be extended to general m, and a connection to a real analogue of Smale's 17 th Problem.