Valuative trees over valued fields
Maria Alberich-Carrami\~nana, Jordi Gu\`ardia, Enric Nart, Joaquim, Ro\'e

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 and a given extension of ordered groups, we analyze the structure of the tree formed by all -valued extensions of to the polynomial ring . As an application, we find a model for the tree of all equivalence classes of valuations on (without fixing their value group), whose restriction to is equivalent to . In the henselian case, we apply these results to show that there is a complete parallelism between the arithmetic properties of irreducible polynomials , encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations on , encoded by their MacLane-Vaqui\'e chains. This parallelism was only known for defectless irreducible polynomials.
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Taxonomy
TopicsRings, Modules, and Algebras · Algebraic Geometry and Number Theory · Advanced Topology and Set Theory
