On Solving Fewnomials Over Intervals in Fewnomial Time
J. Maurice Rojas, Yinyu Ye
备注：
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
设 f 是一个次数为 D、实系数且恰好包含 m 个单项式的单变量多项式。我们证明，在特殊情形 m=3 下，仅使用 O(log(D)log(Dlog(R/eps))) 次算术运算，即可在区间 [0,R] 内以 eps 精度逼近 f 的所有根。特别地，仅使用 O(log^2 D) 次算术运算，即可计算任意有界区间内的根的个数。我们的加速效果显著且近乎最优：此前这两个问题在渐近意义上最紧的复杂度上界均为关于 D 的超线性，而我们的算法复杂度接近各自相应的复杂度下界。我们还讨论了将算法推广到一般 m 的条件，以及其与 Smale 第 17 问题实模拟的联系。
英文摘要：
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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Department of Mathematics and Statistics, University at Albany, State University of New York（数学与统计学系，阿尔巴尼大学，纽约州立大学）；Department of Mathematics, Dartmouth College（数学系，达特茅斯学院）
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INRIA-Rennes, IRMAR and ENS Bruz（法国国家信息与自动化研究所雷恩分中心，雷恩1大学数学与算法研究实验室，布鲁兹高等师范学院）；Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien（沃尔夫冈·泡利研究所（依托维也纳大学数学系），维也纳大学）；IRMAR, Université de Rennes 1 and INRIA-Rennes（雷恩1大学数学与算法研究实验室，法国国家信息与自动化研究所雷恩分中心）
