Some factorization results for formal power series

TL;DR

This paper presents new factorization results and irreducibility criteria for formal power series over principal ideal domains, utilizing prime factorization and Newton polygon theory to extend classical irreducibility tests.

Contribution

It introduces sharp bounds on irreducible factors and extends Dumas's irreducibility criterion to formal power series over discrete valuation domains.

Findings

Derived bounds on the number of irreducible factors.

Established irreducibility criteria based on prime factorization.

Extended classical irreducibility criteria using Newton polygons.

Abstract

In this paper, we obtain some factorization results on formal power series over principle ideal domains with sharp bounds on number of irreducible factors. These factorization results correspondingly lead to irreducibility criteria for formal power series. The information about prime factorization of the constant term up to a unit and that of some higher order terms is utilized for the purpose. Further, using theory of Newton polygons for power series, we extend the classical Dumas irreducibility criterion to formal power series over discrete valuation domains, which in particular, yields several irreducibility criteria.