---
_id: '568'
abstract:
- lang: eng
  text: 'We study robust properties of zero sets of continuous maps f: X → ℝn. Formally,
    we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets
    of all continuous maps g closer to f than r in the max-norm. All of these sets
    are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined
    by A and an element of a certain cohomotopy group which (by a recent result) is
    computable whenever the dimension of X is at most 2n - 3. By considering all r
    > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a
    structure leading to persistence diagrams as in the case of persistent homology
    or well groups. Eventually, we get a descriptor of persistent robust properties
    of zero sets that has better descriptive power (Theorem A) and better computability
    status (Theorem B) than the established well diagrams. Moreover, if we endow every
    point of each zero set with gradients of the perturbation, the robust description
    of the zero sets by elements of cohomotopy groups is in some sense the best possible
    (Theorem C).'
author:
- first_name: Peter
  full_name: Franek, Peter
  id: 473294AE-F248-11E8-B48F-1D18A9856A87
  last_name: Franek
- first_name: Marek
  full_name: Krcál, Marek
  id: 33E21118-F248-11E8-B48F-1D18A9856A87
  last_name: Krcál
citation:
  ama: Franek P, Krcál M. Persistence of zero sets. <i>Homology, Homotopy and Applications</i>.
    2017;19(2):313-342. doi:<a href="https://doi.org/10.4310/HHA.2017.v19.n2.a16">10.4310/HHA.2017.v19.n2.a16</a>
  apa: Franek, P., &#38; Krcál, M. (2017). Persistence of zero sets. <i>Homology,
    Homotopy and Applications</i>. International Press. <a href="https://doi.org/10.4310/HHA.2017.v19.n2.a16">https://doi.org/10.4310/HHA.2017.v19.n2.a16</a>
  chicago: Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” <i>Homology,
    Homotopy and Applications</i>. International Press, 2017. <a href="https://doi.org/10.4310/HHA.2017.v19.n2.a16">https://doi.org/10.4310/HHA.2017.v19.n2.a16</a>.
  ieee: P. Franek and M. Krcál, “Persistence of zero sets,” <i>Homology, Homotopy
    and Applications</i>, vol. 19, no. 2. International Press, pp. 313–342, 2017.
  ista: Franek P, Krcál M. 2017. Persistence of zero sets. Homology, Homotopy and
    Applications. 19(2), 313–342.
  mla: Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” <i>Homology, Homotopy
    and Applications</i>, vol. 19, no. 2, International Press, 2017, pp. 313–42, doi:<a
    href="https://doi.org/10.4310/HHA.2017.v19.n2.a16">10.4310/HHA.2017.v19.n2.a16</a>.
  short: P. Franek, M. Krcál, Homology, Homotopy and Applications 19 (2017) 313–342.
date_created: 2018-12-11T11:47:14Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2021-01-12T08:03:12Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.4310/HHA.2017.v19.n2.a16
ec_funded: 1
intvolume: '        19'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1507.04310
month: '01'
oa: 1
oa_version: Submitted Version
page: 313 - 342
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
- _id: 2590DB08-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '701309'
  name: Atomic-Resolution Structures of Mitochondrial Respiratory Chain Supercomplexes
    (H2020)
publication: Homology, Homotopy and Applications
publication_identifier:
  issn:
  - '15320073'
publication_status: published
publisher: International Press
publist_id: '7246'
quality_controlled: '1'
scopus_import: 1
status: public
title: Persistence of zero sets
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2017'
...