Displaying similar documents to “On the Poincaré series for a plane divisorial valuation.”

Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas, J. Madden, D. Schaub, M. Spivakovsky (2012)

Annales de la faculté des sciences de Toulouse Mathématiques


In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring A . We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second –...

On proximity relations for valuations dominating a two-dimensional regular local ring.

José J. Aparicio, Angel Granja, Tomás Sánchez-Giralda (1999)

Revista Matemática Iberoamericana


The purpose of this paper is to define a new numerical invariant of valuations centered in a regular two-dimensional regular local ring. For this, we define a sequence of non-negative rational numbers δ = {δ(j)} which is determined by the proximity relations of the successive quadratic transformations at the points determined by a valuation ν. This sequence is characterized by seven combinatorial properties, so that any sequence of non-negative rational numbers having the above properties...

One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Veronique Lierde (2011)

Open Mathematics


Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate...