MacLane-Vaquié chains of valuations on a polynomial ring

Abstract

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaquié on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaquié 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-Vaquié 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 $v$.

References in corpus (1)

• Valuations in algebraic field extensions

Cited by in corpus (5)

• Rigidity of valuative trees under henselization
• Invariants of limit key polynomials
• On truncations of valuations
• Small extensions of abelian ordered groups
• Valuations on $K[x]$ approaching a fixed irreducible polynomial