Residual ideals of MacLane valuations

TL;DR

This paper advances the understanding of valuations on polynomial fields by analyzing residual ideals in graded algebras, leading to a detailed structure theorem and applications in classifying irreducible polynomials over local fields.

Contribution

It extends Vaquié's approach by studying residual ideals, providing a structural description of graded algebras of valuations, and linking valuations to polynomial classification.

Findings

Determined the structure of graded algebras for discrete valuations on $K(x)$.

Connected residual ideals to residual polynomials in valuation theory.

Provided a parameterization of irreducible polynomials over local fields up to Okutsu equivalence.

Abstract

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaqui\'e, by using the graded algebra of a valuation. We extend Vaqui\'e's approach by studying residual ideals of the graded algebra of a valuation as an abstract counterpart of certain residual polynomials which play a key role in the computational applications of the theory. As a consequence, we determine the structure of the graded algebra of the discrete valuations on $K(x)$ and we show how these valuations may be used to parameterize irreducible polynomials over local fields up to Okutsu equivalence.