Motivic integration and Milnor fiber

We put forward a uniform narrative that weaves together several variants of Hrushovski– Kazhdan style integral, and describe how it can facilitate the understanding Denef–Loeser motivic Milnor fiber closely related objects. Our study focuses on so-called “nonarchimedean fiber” was introduced by Hrushovski Loeser, our thesis is richer embodiment underlying philosophy construction. The said first developed in more natural complex environment, then extended to real one via descent. In process doing so, we are able provide illuminating new proofs, free resolution singularities, few pivotal results literature, both real. To begin with, zeta function shown be rational, which yields fiber; this an analogue Hrushovski–Loeser Then, applying $T$-convex integration after descent, matching Euler characteristics topological becomes matter simple computation, not only singularities as proof, but also other sophisticated algebro-geometric machineries. Finally, establish, much intuitive manner, Thom–Sebastiani formula, specialized given Guibert–Loeser–Merle.