Generalized Loose Edge Factorization Theorems

Abstract

We extend a factorization theorem by Gwoździewicz 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 Weierstraß polynomials with coefficients in $R$, which generalizes an earlier result by Rond and the author.

10 pages; corrected typos and clarified Lemma 3.10 and adapted the lemmas for the proof slightly