Symmetries of some motivic integrals

TL;DR

This paper derives explicit formulas for certain motivic integrals related to the Milnor number, reveals an unexpected symmetry under parameter inversion, and explores generalized recurrence relations with additional symmetries and differential equations.

Contribution

It provides explicit formulas and uncovers a novel invariance property of motivic integrals, along with a generalized system of recurrence relations exhibiting symmetry.

Findings

Motivic integrals are rational functions of parameters and the affine line class.

An invariance property under simultaneous inversion of parameters and affine line class.

Generalized recurrence relations lead to symmetric solutions satisfying differential equations.

Abstract

We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational functions of the parameters and the class of the affine line. Using a set of natural recurrence relations between them, we prove an unexpected invariance property with respect to the simultaneous inversion of the parameters and the class of the affine line. We also discuss a generalization of this system of recurrence relations whose solutions are also symmetric and satisfy additional differential equations.