Numerically computing real points on algebraic sets

TL;DR

This paper develops a homotopy-based algorithm to compute at least one real point on each connected component of a real algebraic set, improving the ability to determine real roots of polynomial systems.

Contribution

It transforms a classical optimization approach into a homotopy method for real root computation, offering a parallelizable solution for algebraic sets.

Findings

Effective in computing real points on algebraic sets

Parallelizable homotopy approach demonstrated

Examples show practical applicability

Abstract

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.