Local motivic invariants of rational functions in two variables

TL;DR

This paper computes local motivic invariants of rational functions in two variables at indeterminacy points using Newton polygons, linking motivic Milnor fibers to topological and motivic bifurcation sets.

Contribution

It introduces a method to compute motivic Milnor fibers of rational functions in two variables via Newton polygons without non-degeneracy conditions.

Findings

Motivic Milnor fibers are expressed in terms of motives associated with Newton polygon faces.

Under certain smoothness and independence conditions, topological and motivic bifurcation sets coincide.

Both bifurcation sets can be computed from the Newton algorithm.

Abstract

Let $P$ and $Q$ be two polynomials in two variables with coefficients in an algebraic closed field of characteristic zero. We consider the rational function $f=P/Q$. For an indeterminacy point $\text{x}$ of $f$ and a value $c$, we compute the motivic Milnor fiber $S_{f,\text{x}, c}$ in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithms of $P-cQ$ and $Q$ at $\text{x}$, without any condition of non-degeneracy or convenience. In the complex setting, assuming for any $(a,b)\in \mathbb{C}^2$ that $\text{x}$ is a smooth or an isolated critical point of $aP+bQ$, and the curves $P=0$ and $Q=0$ do not have common irreducible component, we prove that the topological bifurcation set $\mathscr{B}_{f,\text{x}}^{\text{top}}$ is equal to the motivic bifurcation set $\mathscr{B}_{f,\text{x}}^{\text{mot}}$ and they are computed from the Newton algorithm.