Factoring formal power series over principal ideal domains

TL;DR

This paper introduces an irreducibility test and factoring algorithm for formal power series over principal ideal domains, extending classical results with a generalized Weierstrass preparation theorem.

Contribution

It provides the first comprehensive irreducibility criteria and factoring methods for formal power series over PIDs, and classifies quotient rings of these series.

Findings

Developed an irreducibility test for formal power series over PIDs.

Created a factoring algorithm with specific qualifications.

Classified all quotient rings of formal power series over PIDs.

Abstract

We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising as quotient rings of $R[[X]]$. Our main tool is a generalization of the $p$-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.