Famille admise associ\'ee \`a une valuation de K(X)

TL;DR

This paper investigates valuations on a transcendental extension of a field and demonstrates that the associated family of valuations does not depend on the choice of generator, highlighting an intrinsic property of these valuations.

Contribution

It proves that the family of valuations associated to an extension valuation is independent of the chosen generator in the polynomial ring.

Findings

The associated family converges towards the valuation μ.

The family of valuations does not depend on the chosen generator x.

The structure of the polynomial ring influences the valuation family.

Abstract

Let K be a field with a valuation $\nu$ and let L = K(x) be a transcendental extension of K, then any valuation $\mu$ of L which extends $\nu$ is determined by its restriction to the polynomial ring K[x]. We know how to associate to this valuation $\mu$ a family of valuations A = ($\mu$i)i$\in$I of K[x], called the associated admise family, which converges in a certain sense towards the valuation $\mu$. Although the definition of this family, as well as the notion of convergence, essentially imply the structure of the polynomial ring, in particular the degree of polynomials, we show in this note that the family A of valuations of L do not depend on the chosen generator x.