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In this paper we use motivic integration and non-archimedean analyticgeometry to study the singularities at infinity of the fibers of a polynomialmap $f\colon \mathbb A^d_\mathbb C \to \mathbb A^1_\mathbb C$. We show that themotive $S_{f,a}^{\infty}$ of the motivic nearby cycles at infinity of $f$ for avalue $a$ is a motivic generalization of the classical invariant$\lambda_f(a)$, an integer that measures a lack of equisingularity at infinityin the fiber $f^{-1}(a)$. We then introduce a non-archimedean analytic nearbyfiber at infinity $\mathcal F_{f,a}^{\infty}$ whose motivic volume recovers themotive $S_{f,a}^{\infty}$. With each of $S_{f,a}^{\infty}$ and $\mathcalF_{f,a}^{\infty}$ can be naturally associated a bifurcation set; we show thatthe first one always contains the second one, and that both contain theclassical topological bifurcation set of $f$ whenever $f$ has isolatedsingularities at infinity.

Motivic and analytic nearby fibers at infinity and bifurcation sets