Persistence of zero sets

We study robust properties of zero sets of continuous maps $f : X \to \mathbb{R}^n$. Formally, we analyze the family $Z_{\lt r} (f) := \lbrace g^{-1} (0) : \lVert g - f \rVert \lt r \rbrace$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. All of these sets are outside $A := \lbrace x : \lvert f(x) \rvert \geqslant r \rbrace$ and we claim that $Z_{\lt r} (f)$ 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 $r \gt 0$ 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).

Keywords

system of equations, computational homotopy theory, cohomotopy group

2010 Mathematics Subject Classification

55-04, 65G20, 68Q25

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The research leading to these results has received funding from Austrian Science Fund (FWF): M 1980, the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007–2013) under REA grant agreement number [291734] and from the Czech Science Foundation (GACR) grant number 15-14484S with institutional support RVO:67985807. The research of Marek Krčál was supported by the project number GACR 17-09142S of the Czech Science Foundation. We are grateful to Sergey Avvakumov, Ulrich Bauer, Marek Filakovský, Amit Patel, Lukáš Vokřínek and Ryan Budney for useful discussions and hints.

Received 26 May 2016

Received revised 25 January 2017

Published 29 November 2017