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 $\text{x}$ 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.