Susanne Müller. Source: Journal of Commutative Algebra, Volume 12, Number 4, 559--572.Abstract:
The aim of this paper is to give a connection between the [math] -pure threshold of a polynomial and the height of the corresponding Artin–Mazur formal group. For this, we consider a quasihomogeneous polynomial [math] of degree [math] equal to the degree of [math] and show that the [math] -pure threshold of the reduction [math] is equal to the log-canonical threshold of [math] if and only if the height of the Artin–Mazur formal group associated to [math] , where [math] is the hypersurface given by [math] , is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree greater than [math] . Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the [math] -pure threshold of a quasihomogeneous polynomial of degree [math] cannot be characterized by the height.