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

Abstract

Let I be an arbitrary ideal in 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 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 I. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.

Newton transformations and motivic invariants at infinity of plane curves

“…We extend these definition to c = ∞, considering the rational function Q/P and defining E ∞ as the fan E 0 for Q/P . We recall the Newton algorithm and refer to [40,1,6,7,4] for more details. Definition 1.6.…”

“…In [31], the second author proved that the set of values c such that S f,x,c ̸ = 0 is a finite set, denoted by B mot f,x and called motivic bifurcation set (Definition 4.7). In this article, similarly to [4,5], assuming k algebraically closed, we investigate the case d = 2 in full generality (namely without any assumptions of convenience or non degeneracy w.r.t any Newton polygon) using ideas of Guibert in [17], Guibert-Loeser-Merle in [18,20], and the works of the first author and Veys in the case of an ideal of k [[x, y]] in [6,7].…”

Newton Transformations and the Motivic Milnor Fiber of a Plane Curve

“…In this article we investigate the case of polynomials in k[x, y] in full generality (namely without any assumptions of convenience or non degeneracy w.r.t any Newton polygon) using ideas of Guibert in [16], Guibert, Loeser and Merle in [17], and the works of the first author and Veys in the case of an ideal of k[[x, y]] in [8,9] (see also [7] for the equivariant case). -Let E be a nonempty finite subset of N 2 .…”