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

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.