On Solving Fewnomials Over Intervals in Fewnomial Time

Details On Solving Fewnomials Over Intervals in Fewnomial Time On Solving Fewnomials Over Intervals in Fewnomial Time

2001-06-27

arxiv.org/abs/math/0106225v2

Mathematics

Numerical Analysis

19 pages, 1 encapsulated postscript figure. Major revision correcting many typos and minor errors. Additional discussion on co

Scientific paper

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.

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