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arXiv · 1911.01714

MacLane-Vaqui\'e chains of valuations on a polynomial ring

Enric Nart
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Abstract

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaqui\'e chains of residually transcendental valuations, and we prove that every valuation $\mu$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaqui\'e chain. This chain underlying $\mu$ is essentially unique and contains arithmetic data yielding an explicit description of the graded algebra of $\mu$ as an algebra over the graded algebra of $v$.

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Enric Nart. 2019-11-05. MacLane-Vaqui\'e chains of valuations on a polynomial ring. https://doi.org/10.2140/pjm.2021.311.165

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