2014/06/03 by Govantes, F. J. Herrera, Mahboub, W., Acosta, M. A. Olalla +1
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1406.0657
Let ι:K\hookrightarrow L≅ K(x) be a simple transcendental extension of valued fields, where K is equipped with a valuation ν of rank 1. That is, we assume given a rank 1 valuation ν of K and its extension ν' to L. Let (Rν,Mν,kν) denote the valuation ring of ν. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaquié, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to ι a countable well ordered set Q=\Qi\i∈Λ⊂ K[x]; the Qi are called \bf key polynomials. Key polynomials Qi which have no immediate predecessor are called \bf limit key polynomials. Let βi=ν'(Qi). We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if char kν=0 then the set of key polynomials has order type at most ω, while in the case char kν=p>0 this order type is bounded above by ω×ω, where ω stands for the first infinite ordinal.