The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold

Abstract

AbstractLet${\mathcal I}$be an arbitrary ideal in${\mathbb C}$[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to${\mathcal I}$, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of${\mathcal I}$. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.