Title: Removing Isolated Zeroes by Homotopy,
Abstract: Suppose that the inverse image of the zero vector by a continuous map
$f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local
obstruction to removing this isolated zero by a small perturbation,
generalizing the notion of index for vector fields, the $q=n$ case. The
existence of a continuous map $g$ which approximates $f$ but is nonvanishing
near $P$ is equivalent to a topological property we call "locally inessential,"
and for dimensions $n$, $q$ where $\pi_{n-1}(S^{q-1})$ is trivial, every
isolated zero is locally inessential. We consider the problem of constructing
such an approximation $g$, and show that there exists a continuous homotopy
from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic
map, then there exists such a homotopy which is also semialgebraic. For $q=2$
and $f$ real analytic with a locally inessential isolated zero, 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.