On the worst-case arithmetic complexity of approximating zeros of systems of polynomials
SIAM Journal on Computing
Complexity of Bezout's theorem III: condition number and packing
Journal of Complexity - Festschrift for Joseph F. Traub, Part 1
Complexity of Bezout's theorem V: polynomial time
Selected papers of the workshop on Continuous algorithms and complexity
Complexity of Bezout's theorem IV: probability of success; extensions
SIAM Journal on Numerical Analysis
Complexity and real computation
Complexity and real computation
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
SIAM Journal on Matrix Analysis and Applications
On Smale's 17th Problem: A Probabilistic Positive Solution
Foundations of Computational Mathematics
Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric
Foundations of Computational Mathematics
Complexity of Bezout’s Theorem VII: Distance Estimates in the Condition Metric
Foundations of Computational Mathematics
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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.