Tame fields, Graded Rings and Finite Complete Sequences of Key Polynomials
Caio Henrique Silva de Souza
TL;DR
This work develops a precise criterion linking tameness of a valued field to finite complete sequences of Mac Lane-Vaquié key polynomials and the behavior of associated graded rings. By analyzing augmentations, MLV chains, and defect formulas, it shows that a valued field $(K,v)$ is tame if and only if $vK$ is $p$-divisible, $Kv$ is perfect, and every simple algebraic extension admits a finite complete sequence of key polynomials; this is interpreted through the surjectivity of the Frobenius on the associated graded ring. The results connect tameness with the structure of ${ m gr}(K)$, algebraic maximality, and defect phenomena, and extend to simply defectless and purely inertial/ramified extensions. The paper also develops a robust framework for using complete sequences of key polynomials to study local uniformization in positive characteristic, with broad implications for valuation theory and ramification. Overall, it provides a concrete, checkable criterion for tameness that ties graded-ring properties to key-polynomial data and defect, offering a path toward refined control of extensions in tame fields.
Abstract
In this paper, we present a criterion for $(K,v)$ to be henselian and defectless in terms of finite complete sequences of key polynomials. For this, we use the theory of Mac Lane-Vaquié chains and abstract key polynomials. We then prove that a valued field $(K,v)$ is tame if and only if $vK$ is $p$-divisible, $Kv$ is perfect and every simple algebraic extension of $K$ admits a finite complete sequence of key polynomials. The properties $vK$ $p$-divisible and $Kv$ perfect are described by the Frobenius endomorphism on the associated graded ring. We also make considerations on simply defectless and algebraically maximal valued fields and purely inertial and purely ramified extensions.