Some variational properties of tangent directions at infinity of real algebraic sets

TL;DR

This paper explores the relationship between asymptotic critical values of polynomial functions and tangent directions at infinity, providing conditions for equisingularity at infinity of polynomial fibers.

Contribution

It introduces a novel connection between asymptotic critical values and tangent directions at infinity, offering new necessary conditions for equisingularity at infinity.

Findings

Relates asymptotic critical values to tangent directions at infinity.

Provides necessary conditions for equisingularity at infinity.

Links volume functions with tangent directions at infinity.

Abstract

In this paper, we relate the set of asymptotic critical values of a polynomial function $f$ with the set of discontinuity of two functions, the multivalued function which associate to each value $t$ the set of tangent directions at infinity of the fiber $f^{-1}(t)$ and the composition of the $(n-2)$-dimensional volume function with the first one. This gives necessary conditions of equisingularity at infinity for the family of the fibers of a real polynomial function.