Condition length and complexity for the solution of polynomial systems

TL;DR

This paper advances the understanding of solving polynomial systems by improving complexity bounds and condition number estimates, using homotopy methods to simplify previous approaches and achieve polynomial-time solutions.

Contribution

It provides new complexity bounds and condition number estimates for polynomial system solutions, replacing elimination theory with homotopy methods for improved efficiency.

Findings

Improved complexity bounds for randomized algorithms solving polynomial systems.

Enhanced estimates of the average condition number for polynomial systems.

Simplified proof of the main complexity result using homotopy methods.

Abstract

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale 2000). The main progress on Smale's problem is Beltr\'an-Pardo (2011) and B\"urgisser-Cucker (2010). In this paper we will improve on both approaches and we prove an important intermediate result. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltr\'an-Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in B\"urgisser-Cucker (2010), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last Theorem is the main result of B\"urgisser-Cucker (2010). We give a proof of it relying only on homotopy methods, thus removing the need for the elimination theory methods used in B\"urgisser-Cucker (2010). We build on methods developed in Armentano et al. (2015).