Okutsu invariants and Newton polygons

TL;DR

This paper characterizes Okutsu invariants and families using Newton polygons of higher order, providing formulas, new invariants, and applications to Montes' algorithm for analyzing irreducible polynomials over local fields.

Contribution

It introduces a new characterization of Okutsu families via Newton polygons and expands the applications of Montes' algorithm in polynomial analysis.

Findings

Closed formulas for Okutsu invariants

Discovery of new Okutsu invariants

Construction of Montes approximations

Abstract

Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in terms of certain Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to F(x); these are monic irreducible polynomials sufficiently close to F(x) to share all its Okutsu invariants. This perspective widens the scope of applications of Montes' algorithm, which can be reinterpreted as a tool to compute the Okutsu polynomials and a Montes approximation, for each irreducible factor of a monic separable polynomial f(x) in O[x].