Urysohn in action: separating semialgebraic sets by polynomials

TL;DR

This paper develops an algorithm in real algebraic geometry to constructively find a polynomial separator between two disjoint compact semialgebraic sets, inspired by Uryshon's lemma.

Contribution

It introduces a novel algorithm that efficiently computes separating polynomials for disjoint semialgebraic sets, bridging topology and algebraic geometry.

Findings

Algorithm successfully computes separating polynomials in practical cases

Provides a constructive approach to Uryshon's lemma in algebraic geometry

Enhances methods for separating semialgebraic sets computationally

Abstract

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive and efficient in the context of real algebraic geometry. Namely, given two compact disjoint basic semialgebraic sets which are contained in an $n$-dimensional box, we provide an algorithm that computes a separating polynomial greater than or equal to 1 on the first set and less than or equal to 0 on the second one.