Smale’s 17th Problem: I

Abstract

In 1998, at the request of the International Mathematical Union, Steve Smale published a list of mathematical problems for the twenty-first century. The 17th problem in the list reads as follows: Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? Using the framework developed in the previous chapters, it is easy to give precise meaning to the terms “approximately”, “average”, and “uniform algorithm”. However, as of today, there is no conclusive answer to this problem. But a number of partial results towards such an answer have been obtained in recent years. We devote this and the next chapter to the exposition of these results. The core of this is an algorithm, proposed by Carlos Beltrán and Luis Miguel Pardo, that finds an approximate zero in average polynomial time but makes random choices (flips coins, so to speak) during the computation. This adaptive homotopy algorithm with random start system leads to a probabilistic solution of Smale’s 17th problem.KeywordsRiemannian ManifoldDiscriminant VarietyStandard DistributionNaive AlgorithmCoarea FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.