# Loose edges and factorization theorems

## TL;DR

This paper investigates the factorization properties of elements in regular local rings with specific geometric features in their Newton polyhedra, establishing conditions under which such elements factorize in the ring's completion.

## Contribution

It introduces a new factorization criterion based on loose edges of Newton polyhedra and symbolic restrictions, extending understanding of factorization in regular local rings.

## Key findings

- Factorization in the m-adic completion when the Newton polyhedron has a loose edge.
- Coprimality of polynomials on the edge implies factorization.
- Conditions linking geometric properties of Newton polyhedra to algebraic factorization.

## Abstract

Let $ R $ be a regular local ring with maximal ideal $ \mathfrak{m} $. We consider elements $ f \in R $ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two coprime polynomials, then $f$ factorizes in the $ \mathfrak{m} $-adic completion.