Offset Hypersurfaces and Persistent Homology of Algebraic Varieties

In this paper, we study the true persistent homology 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 (Euclidean Distance Degree). 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.