Urysohn in action: separating semialgebraic sets by polynomials

2022/07/01 by Korda, Milan, Lasserre, Jean-Bernard, Lazarev, Alexey +2
#Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2207.00570

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.

Related

• Separation of Semialgebraic Sets
• Continuity and Algebraic Structure of the Urysohn space
• Minimizing Polynomials Over Semialgebraic Sets
• Decomposition of sets in real algebraic geometry
• On the minimum of a polynomial function on a basic closed semialgebraic set and applications