Description

• Title:
  Removing isolated zeroes by homotopy
• Creator:
  Coffman, Adam;
  Lebl, Jiří
• Description:
  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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.07.13
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 54, No 1 (September 2019); 275 - 296;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.042/18015 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.042/18016
• Subject:
  Isolated zero;
  semialgebraic map;
  singularities of differentiable mappings;
  Primary;
  57R45;
  Secondary;
  14P10;
  31B25;
  35J25;
  41A29;
  57R25;
  58K25;
  58K45;
  58K60
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.042
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other