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On Solving Fewnomials Over Intervals in Fewnomial Time

J. Maurice Rojas·Yinyu Ye
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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 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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J. Maurice Rojas, Yinyu Ye. 2004-03-04. On Solving Fewnomials Over Intervals in Fewnomial Time. https://arxiv.org/abs/math/0106225

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