On Solving Fewnomials Over Intervals in Fewnomial 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 eps all the roots of f in the interval [0,R] using just O(log(D)log(Dlog(R/eps))) 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.
Cite
@article{arxiv.math/0106225,
title = {On Solving Fewnomials Over Intervals in Fewnomial Time},
author = {J. Maurice Rojas and Yinyu Ye},
journal= {arXiv preprint arXiv:math/0106225},
year = {2025}
}
Comments
19 pages, 1 encapsulated postscript figure. Major revision correcting many typos and minor errors. Additional discussion on connection to Smale's 17th Problem and some new references are included