NDLI: Solving polynomial equations in smoothed polynomial time and a near solution to smale's 17th problem

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Solving polynomial equations in smoothed polynomial time and a near solution to smale's 17th problem

Content Provider	ACM Digital Library
Author	Bürgisser, Peter Cucker, Felipe
Abstract	The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in \$n\$ unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltran and Luis Miguel Pardo who exhibited a randomized algorithm, call it LV, doing so. In this paper we further extend this result in several directions. Firstly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and $σ^{-1},$ where σ controls the size of the random perturbation of the input systems. Secondly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f, of the expected running time of LV with input f. In addition to its dependence on N this bound also depends on the condition of f. Thirdly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is NO(log log N). This is nearly a solution to Smale's 17th problem.
Starting Page	503
Ending Page	512
Page Count	10
File Format	PDF
ISBN	9781450300506
DOI	10.1145/1806689.1806759
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2010-06-05
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Smoothed analysis Approximate zero Homotopy methods Complexity Polynomial equation solving Polynomial time
Content Type	Text
Resource Type	Article

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