Newton transformations and motivic invariants at infinity of plane curves

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 x of f and a value c, we compute the motivic Milnor fiber S f,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 x, without any condition of non-degeneracy or convenience. In the complex setting, assuming for any (a, b) ∈ C 2 that x is 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 B top f,x is equal to the motivic bifurcation set B mot f,x and they are computed from the Newton algorithm.