Offset Hypersurfaces and Persistent Homology of Algebraic Varieties

TL;DR

This paper explores the algebraic and topological properties of offset hypersurfaces of algebraic varieties, linking persistent homology with algebraic optimization and providing new computational insights.

Contribution

It proves algebraicity of key quantities in persistent homology, relates offset hypersurfaces to Euclidean Distance Degree, and characterizes topologically interesting points.

Findings

Proves algebraicity of quantities in persistent homology.

Expresses offset hypersurface degree via Euclidean Distance Degree.

Describes the non-properness locus and topologically relevant points.

Abstract

In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and algebraic optimization. Namely, we express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean Distance Degree of the starting variety, obtaining a new way to compute these degrees. Finally, we describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.