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

It is shown that the auto-Igusa zeta function of a plane curve singularity is rational. This gives a new criterion for a plane curve over an algebraically closed field of characteristic zero to be smooth at a point.Comment: 9 page

Motivic integration over nilpotent structures

This thesis concerns developing the notion of Motivic Integration in such a way that it captures infinitesimal information yet reduces to the classical notion of motivic integration for reduced schemes. Moreover, I extend the notion of Motivic Integration from a discrete valuation ring to any complete Noetherian ring with residue field $\kappa$, where $\kappa$ is any field. Schoutens\u27 functorial approach (as opposed to the traditional model theoretic approach) allows for some very general notions of motivic integration. However, the central focus is on using this general framework to study generically smooth schemes, then non-reduced schemes, and then, finally, formal schemes. Finally, a computational approach via Sage for computing the equations defining affine arc spaces is introduced and implemented