Local motivic invariants of rational functions in two variables

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 Sf,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)∈ C² 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 Bf,xᵗᵒᵖ is equal to the motivic bifurcation set Bf,xᵐᵒᵗ and they are computed from the Newton algorithm.