Connectivity in Semi-Algebraic Sets I

TL;DR

This paper introduces a symbolic method based on gradient ascent to decide connectivity between points in semi-algebraic sets, providing proofs of correctness and demonstrating effectiveness through examples.

Contribution

It presents the symbolic component of a new gradient ascent-based approach for connectivity analysis in semi-algebraic sets, with proofs and examples.

Findings

Method is correct and terminates as proven.

Effective in analyzing non-trivial examples.

Lays groundwork for the subsequent numeric part.

Abstract

A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two given points in a semi-algebraic set are connected; that is, whether the two points lie in the same connected component. In particular, we consider the semi-algebraic set defined by f <> 0 where f is a given polynomial with integer coefficients. The motivation comes from the observation that many important or non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to its importance, there has been intense research effort on the problem. We will describe a symbolic-numeric method based on gradient ascent. The method will be described in two papers. The first paper (the present one) will describe the symbolic part and the forthcoming second paper will describe the numeric part. In the present paper, we give proofs of correctness and termination for the symbolic part and illustrate the efficacy of the method using several non-trivial examples.