Atypical values at infinity of real polynomial maps with $2$-dimensional fibers

TL;DR

This paper characterizes atypical values at infinity for certain real polynomial maps with three variables, extending previous results for two-variable cases, and provides criteria based on gradient vector field indices.

Contribution

It generalizes the characterization of atypical values at infinity from two-variable to three-variable polynomial maps with 2-dimensional fibers.

Findings

Atypical values at infinity are characterized by sum of gradient indices.

Provides a new criterion for identifying atypical values at infinity.

Extends known results to higher-dimensional fibers.

Abstract

We characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is an analogy of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are $2$-dimensional surfaces.