A Faster Solution to Smale's 17th Problem I: Real Binomial Systems.
Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation(2019)
摘要
Suppose F:=(f_1,łdots,f_n) is a system of random n-variate polynomials with f_i having degree łeq\!d_i and the coefficient of x^a_1 _1\cdots x^a_n _n in f_i being an independent complex Gaussian of mean 0 and variance \fracd_i! a_1!\cdots a_n!łeft(d_i-\sum^n_j=1 a_j \right)! . Recent progress on Smale's 17þth Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, Bü rgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of F using just N^O(1) arithmetic operations on average, where N\!:=\!\sum^n_i=1 \frac(n+d_i)! n!d_i! (=n(n+\max_i d_i)^O(\min\n,\max_i d_i)\ ) is the maximum possible total number of monomial terms for such an F. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when F is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just O(n^3łog^2(n\max_i d_i)) arithmetic operations on average. Furthermore, our approach allows real Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in nłog \max_i d_i when F has more terms.
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关键词
approximate root, average-case complexity, newton iteration, random polynomial, real roots, smale's 17th problem, sparse polynomial
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