Loose edges

TL;DR

This paper investigates the factorization properties of multivariable formal power series with specific geometric features in their Newton polyhedron, demonstrating conditions under which such series can be factorized.

Contribution

It establishes a new criterion for factorization of power series based on the structure of a loose edge in their Newton polyhedron and the coprimality of symbolic restrictions.

Findings

Factorization of power series with loose edges in their Newton polyhedron

Condition that symbolic restriction being a product of coprime polynomials implies factorization

Extension of factorization results to power series rings

Abstract

We consider formal power series in several variables with coefficients in arbitrary field such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of the power series $f$ to such an edge is a product of two coprime polynomials, then $f$ factorizes in the ring of power series.