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Topological Methods in Nonlinear Analysis

Removing isolated zeroes by homotopy
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Removing isolated zeroes by homotopy

Authors

  • Adam Coffman https://orcid.org/0000-0002-1437-7525
  • Jiří Lebl https://orcid.org/0000-0002-9320-0823

Keywords

Isolated zero, semialgebraic map, singularities of differentiable mappings

Abstract

Suppose that the inverse image of the zero vector by a continuous map $f\colon {\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call ``local inessentiality of zeros'', generalizing the notion of index zero for vector fields, the $q=n$ case. We consider the problem of constructing such an approximation $g$ and a continuous homotopy $F(x,t)$ from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists $F$ also semialgebraic. If $q=2$ and $f$ is real analytic with a locally inessential zero, then there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

References

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Published

2019-07-13

How to Cite

1.
COFFMAN, Adam and LEBL, Jiří. Removing isolated zeroes by homotopy. Topological Methods in Nonlinear Analysis. Online. 13 July 2019. Vol. 54, no. 1, pp. 275 - 296. [Accessed 7 July 2025].
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