Valuative trees over valued fields

TL;DR

This paper studies the structure of valuative trees over valued fields, analyzing extensions of valuations to polynomial rings, and reveals a parallelism between polynomial arithmetic and valuation properties, extending known results to defectful cases.

Contribution

It introduces a comprehensive model for the tree of valuations on polynomial rings over valued fields, generalizing previous defectless-focused results to include defectful polynomials.

Findings

Constructed a model for the tree of all valuation extensions on K[x]

Established a parallelism between irreducible polynomial properties and valuation chains

Extended valuation-theoretic results to defectful irreducible polynomials

Abstract

For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrow\Lambda$ of ordered groups, we analyze the structure of the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial ring $K[x]$. As an application, we find a model for the tree of all equivalence classes of valuations on $K[x]$ (without fixing their value group), whose restriction to $K$ is equivalent to $v$. In the henselian case, we apply these results to show that there is a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaqui\'e chains. This parallelism was only known for defectless irreducible polynomials.