On sign conditions over real multivariate polynomials

TL;DR

This paper introduces a probabilistic algorithm to efficiently identify points in each connected component of sign conditions over real polynomials, improving complexity bounds and extending to closed sign conditions.

Contribution

It presents a novel probabilistic method for analyzing sign conditions over real polynomials, with improved complexity bounds and applicability to closed sign conditions.

Findings

Algorithm finds points in each connected component of sign conditions.

Complexity bounds are improved over previous methods.

Method extends to closed sign conditions.

Abstract

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.