Let $K$ be a field equipped with a valuation $\nu$ and let $L=K(x)$ be a transcendental extension of $K$. Every 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 $\mathcal{A}={({\mu }_i)}_{i\in I}$ of $K[x]$, called the associated admitted family that converges in a certain sense towards the valuation $\mu$. Although the definition of this family, as well as the notion of convergence, use the structure of the polynomial ring, in particular, the degree of polynomials, we show in this note that the family $\mathcal{A}$ of valuations of $L$ does not depend on the chosen generator $x$.

中文翻译：

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Famille admiseassociéeàunvaluation de K（X）

让 $ķ$ 成为具备估值能力的领域 $\nu$ 然后让 $\mathrm{大号}=ķ（X）$ 超越 $ķ$。每次估值$\mu$ 的 $\mathrm{大号}$ 延伸 $\nu$ 由其对多项式环的限制确定 $ķ[X]$。我们知道如何与该估值相关$\mu$ 估值家族 $\mathcal{一种}={（{\mu }_{\mathrm{一世}}）}_{\mathrm{一世}\in \mathrm{一世}}$ 的 $ķ[X]$，被称为相关接纳家庭，在某种意义上趋向于估值 $\mu$。尽管该族的定义以及收敛的概念都使用了多项式环的结构，尤其是多项式的阶数，但我们在本说明中显示该族$\mathcal{一种}$ 的估值 $\mathrm{大号}$ 不依赖于所选择的发电机 $X$。