## Sampling and homology via bottlenecks

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Sandra Di Rocco, David Eklund and Oliver Gäfvert
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## Abstract:

In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact affine variety. The procedure is partly based on computing*bottlenecks*of the variety. Using geometric information such as the bottlenecks and the

*local reach*we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne

*et al.*[

*Sampling real algebraic varieties for topological data analysis*, arXiv:1802.07716, 2018].

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## Additional Information

**Sandra Di Rocco**- Affiliation: Department of mathematics, KTH, 10044 Stockholm, Sweden
- MR Author ID: 606949
- ORCID: 0000-0002-7186-1524
- Email: dirocco@math.kth.se
**David Eklund**- Affiliation: RISE, Research Institutes of Sweden, Isafjordsgatan 22, 16440 Kista, Sweden
- MR Author ID: 906385
- Email: daek@math.kth.se
**Oliver Gäfvert**- Affiliation: Mathematical Institute, University of Oxford, United Kingdom
- Email: oliver.gafvert@maths.ox.ac.uk
- Received by editor(s): November 2, 2020
- Received by editor(s) in revised form: October 13, 2021, and April 12, 2022
- Published electronically: July 22, 2022
- Additional Notes: The third author was supported by the Thematic Einstein Semester
*”Varieties, Polyhedra, Computation”*by the Berlin Einstein Foundation. All three authors were supported by Vetenstapsrådet grants [NT:2014-4763], [NT:2018-03688] - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2969-2995 - MSC (2020): Primary 13P25, 14Q20, 14P25
- DOI: https://doi.org/10.1090/mcom/3757
- MathSciNet review: 4473110