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Some factorization results for formal power series

Rishu Garg, Jitender Singh

TL;DR

The paper investigates factorization and irreducibility of formal power series over principal ideal domains and discrete valuation domains, seeking sharp bounds on the number of irreducible factors from the prime factorization of the constant term and certain higher coefficients. It develops a suite of results, including an inequality $oldsymbol{oldsymbol{oldomega}(a_0)oldsymbol{oldsymbol{ abla}oldsymbol{ ext{O}}}} obreak obreak obreak obreak ext{≤} obreak oldsymbol{oldsymbol{ ext}{ obreak ext{O}}}_f obreak ext{≤} oldsymbol{oldsymbol{ ext{$ ext{O}$}(a_0)}$ for $f= extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle obreak obreak obreak obreak obreak z^i}$ and corresponding corollaries, together with a DVR generalization that bound the number of irreducible factors by $ ext{min}\{k, ext{ }j+ ext{ } obreak ext{ell} obreak ight ext{ and, crucially, a Newton polygon–based irreducibility criterion (extending Dumas) that yields concrete irreducibility results over $ obreak obreak obreak ext{Z}$, $ obreak obreak Z[ ext{i}]$, and related rings. The results are complemented by explicit examples demonstrating irreducibility and multiplicative factorizations, including extensions to localizations and Gaussian integers, highlighting the practical applicability of the criteria.

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.

Some factorization results for formal power series

TL;DR

The paper investigates factorization and irreducibility of formal power series over principal ideal domains and discrete valuation domains, seeking sharp bounds on the number of irreducible factors from the prime factorization of the constant term and certain higher coefficients. It develops a suite of results, including an inequality ext{O} for and corresponding corollaries, together with a DVR generalization that bound the number of irreducible factors by obreak obreak obreak ext{Z} obreak obreak Z[ ext{i}]$, and related rings. The results are complemented by explicit examples demonstrating irreducibility and multiplicative factorizations, including extensions to localizations and Gaussian integers, highlighting the practical applicability of the criteria.

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.
Paper Structure (3 sections, 17 theorems, 32 equations)

This paper contains 3 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Main results and proofs
  3. Examples

Key Result

Theorem A

Let $R$ be a principle ideal domain. Let $f=\sum_{i=0}^\infty a_iz^i\in R[[z]]$ be such that $a_0$ is a nonzero nonunit. If $a_0$ is a product of two nonassociate elements $m$ and $n$ in $R$, then $f$ is not irreducible in $R[[z]]$.

Theorems & Definitions (34)

  • Theorem A
  • proof : Proof of Theorem \ref{['th:A']}
  • Theorem B
  • Theorem C
  • proof : Proof of Theorem \ref{['th:1b']}
  • Theorem D
  • Theorem 1
  • proof : Proof of Theorem \ref{['th:1']}
  • Corollary 2
  • proof : Proof of Corollary \ref{['c:1']}
  • ...and 24 more