research-article

Offset hypersurfaces and persistent homology of algebraic varieties

Authors:

Madeleine WeinsteinAuthors Info & Claims

Volume 74, Issue C

https://doi.org/10.1016/j.cagd.2019.101767

Published: 01 October 2019 Publication History

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.

Highlights

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The algebraicity of the persistent homology of the offset filtration is proved.

•

Degree of the offset hypersurface is strongly connected to the Euclidean Distance degree.

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The medial axis is a subset of the Euclidean Distance discriminant.

References

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Cited By

La Valle CTonelli-Cueto J(2024)Some Lower Bounds on the Reach of an Algebraic VarietyProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669693(217-225)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669693

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Offset hypersurfaces and persistent homology of algebraic varieties

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cover image Computer Aided Geometric Design

Computer Aided Geometric Design Volume 74, Issue C

Oct 2019

142 pages

ISSN:0167-8396

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Elsevier B.V.

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 October 2019

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Cited By

La Valle CTonelli-Cueto J(2024)Some Lower Bounds on the Reach of an Algebraic VarietyProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669693(217-225)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669693

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