Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

Smale’s 17th Problem: Average Polynomial Time to Compute Affine and Projective Solutions

In the series of papers [SS93a, SS93b, SS93c, SS94, SS96], Shub and Smale defined and studied in depth a homotopy method for solving systems of polynomial equations. Some articles preceding this new treatment were [Kan49, Sma86, Ren87, Kim89, Shu93]. Other authors have also treated this approach in [Mal94, Yak95, Ded97, BCSS98, Ded01, Ded06, MR02] and more recently in [Shu08, BS08]. In a previo...

A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time

We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17 problem. The main idea is to make us...

On average polynomial time

One important reason to consider average case complexity is whether all or at least some signi cant part of NP can be solved e ciently on average for all reasonable distributions Let PP comp be the class of problems which are solvable in average polynomial time for every polynomial time computable input distribution Following the framework of Levin the above question can now be formulated as Is...

Affine and projective planes

A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible decomposition of a variety is typically understood in affine space, including also those components with zero coordinates. We present a polyhedral method to compute ...