The auto igusa-zeta function of a plane curve singularity is rational

We show that the auto Igusa-zeta function ζC,p(t) of a plane curve C over an algebraically closed field k is rational away from points p ? C of wild ramification-i.e., it is of the form f(t)/g(t) where f(t) ε Gr(Var _k )[L ^-1 , t], where Gr(Var _k ) is the Grothendieck ring of varieties, and g(t) = Π ⁿ _i =1(1 - L ^ai t ^bi ) with a _i ε ℤ and b _i ε ℕ \ {0}, where L := [A1/k] is the Leftshetz motive. As a consequence, we give a new characterization for a curve C on a smooth surface S to be smooth at a point p on C when the ground field is algebraically closed and of characteristic zero.