12: Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets

In this chapter, we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map $f:{\mathbb{A}}_{ℂ}^d\to {\mathbb{A}}_{ℂ}^1$. We show that the motive $S_{f,a}^{\infty }$ of the motivic nearby cycles at infinity of f for a value a is a motivic generalization of the classical invariant λ_f(a), an integer that measures a lack of equisingularity at infinity in the fiber f⁻¹(a). We then introduce a non-archimedean analytic nearby fiber at infinity ${ℱ}_{f,a}^{\infty }$ whose motivic volume recovers the motive $S_{f,a}^{\infty }$. With each of $S_{f,a}^{\infty }$ and ${ℱ}_{f,a}^{\infty }$, one can naturally associated a bifurcation set; we show that the first one always contains the second one, and that both contain the classical topological bifurcation set of f if f has isolated singularities at infinity.