Newton transformations and motivic invariants at infinity of plane curves

TL;DR

This paper provides a motivic and topological analysis of polynomial plane curves at infinity, expressing invariants via Newton polygons and introducing algorithms for their computation.

Contribution

It introduces a new expression for motivic invariants at infinity of plane curves using Newton polygons, without non-degeneracy conditions, and relates classical and motivic invariants.

Findings

Derived formulas for motivic Milnor fibers at infinity

Computed Euler characteristic of generic fibers via Newton polygons

Established equality between topological and motivic bifurcation sets

Abstract

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial $f$ in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of $f$ without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of $f$ in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if $f$ has isolated singularities, we compute similarly the classical invariants at infinity $\lambda_{c}(f)$ which measures the non equisingularity at infinity of the fibers of $f$ in $\mathbb P^2$, and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.