Generalized Loose Edge Factorization Theorems

TL;DR

This paper generalizes a factorization theorem for formal power series to complete regular local rings, providing new criteria for factorization based on Newton polyhedra and extending previous results to Weierstrass polynomials.

Contribution

It extends a known factorization theorem from formal power series to complete regular local rings, including a new factorization criterion based on Newton polyhedra.

Findings

Generalizes factorization theorem to complete regular local rings.

Provides a new criterion for factorization using Newton polyhedra.

Extends factorization results to Weierstrass polynomials.

Abstract

We extend a factorization theorem by Gwo\'zdziewicz and Hejmej from the ring of formal power series to any complete regular local ring $ R $. More precisely, let $ f \in R $ and assume that its Newton polyhedron has a loose edge such that the initial formal of $ f $ along the latter is a product of two coprime polynomials, where one of them is not divided by any variable. Then this provides a factorization of $ f $ in $ R $. As a consequence we obtain a factorization theorem for Weierstra{\ss} polynomials with coefficients in $ R $, which generalizes an earlier result by Rond and the author.