Removing isolated zeroes by homotopy (Q2336091)

Suppose that the inverse image \(f^{-1}(0)\) by a continuous map \(f:\mathbb{R}^n \rightarrow\mathbb{R}^q\) 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 called by the authors ``local inessentiality of zeros'', generalizing in the case \(q=n\) the notion of index zero for vector fields. In this paper the authors study the problem of constructing such an approximation \(g\) and a continuous homotopy \(F(x, t)\) from \(f\) to \(g\) through locally nonvanishing maps. The paper is organized into sections as follows : notation, nonvanishing approximation, homotopy through nonvanishing maps, real analytic maps to the plane, polynomial examples. Other papers by the first author directly connected to this topic are [Beitr. Algebra Geom. 43, No. 2, 451--477 (2002; Zbl 1029.32020); Linear Algebra Appl. 370, 41--83 (2003; Zbl 1049.14042)].