Motivic integrals and functional equations

A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived by using the Denef–Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant, and there is a simple recursive algorithm to find its coefficients. The method is fairly universal and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered collections of arcs.