Persistence of zero sets

P. Franek, M. Krcál, Homology, Homotopy and Applications 19 (2017) 313–342.


Journal Article | Published | English
Abstract
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).
Publishing Year
Date Published
2017-01-01
Journal Title
Homology, Homotopy and Applications
Volume
19
Issue
2
Page
313 - 342
ISSN
IST-REx-ID

Cite this

Franek P, Krcál M. Persistence of zero sets. Homology, Homotopy and Applications. 2017;19(2):313-342. doi:10.4310/HHA.2017.v19.n2.a16
Franek, P., & Krcál, M. (2017). Persistence of zero sets. Homology, Homotopy and Applications, 19(2), 313–342. https://doi.org/10.4310/HHA.2017.v19.n2.a16
Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” Homology, Homotopy and Applications 19, no. 2 (2017): 313–42. https://doi.org/10.4310/HHA.2017.v19.n2.a16.
P. Franek and M. Krcál, “Persistence of zero sets,” Homology, Homotopy and Applications, vol. 19, no. 2, pp. 313–342, 2017.
Franek P, Krcál M. 2017. Persistence of zero sets. Homology, Homotopy and Applications. 19(2), 313–342.
Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” Homology, Homotopy and Applications, vol. 19, no. 2, International Press, 2017, pp. 313–42, doi:10.4310/HHA.2017.v19.n2.a16.

Link(s) to Main File(s)
Access Level
OA Open Access

Export

Marked Publications

Open Data IST Research Explorer

Search this title in

Google Scholar