Okutsu frames of irreducible polynomials over henselian fields

TL;DR

This paper establishes a comprehensive connection between the arithmetic properties of irreducible polynomials over henselian fields, captured by Okutsu frames, and the valuation-theoretic properties of their induced valuations, described by MacLane-Vaquié chains.

Contribution

It extends the known parallelism from defectless polynomials to all irreducible polynomials over henselian fields.

Findings

Established a complete parallelism between Okutsu frames and MacLane-Vaquié chains.

Extended the known results from defectless to all irreducible polynomials.

Provides a unified framework linking polynomial arithmetic and valuation theory.

Abstract

For a henselian valued field $(K,v)$ we establish 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.