Offset Hypersurfaces and Persistent Homology of Algebraic Varieties

2018/03/20 by Horobet, Emil, Weinstein, Madeleine · 3 citations
#14H45 #14M12 #14P25 #41A65 #55R80 #68W30 #90C26 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1803.07281

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.

Cited by

• The Hermitian Distance degree of an Algebraic Variety
• Infinite Euclidean Distance Discriminants
• Osculating Geometry and Higher-Order Distance Loci

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