A Goldbach theorem for Laurent series semidomains

TL;DR

This paper develops a Goldbach-type framework for Laurent polynomials and Laurent series over additively reduced, additively atomic semidomains. It shows that the key condition $\mathcal{A}_{+}(S)=S^{\times}$ is equivalent to every non-monomial Laurent polynomial being expressible as a sum of two multiplicative irreducibles, and it further characterizes the precise exceptional forms that resist such decompositions. Extending to Laurent series, the authors prove an analogous three-irreducible decomposition result, and demonstrate that non-polynomial Laurent series admit uncountably many representations as sums of three irreducibles. The results generalize prior work in the area and, in particular, recover and extend known two-irreducible decompositions for multivariate Laurent polynomials over $\mathbb{N}$, while revealing a richly non-unique decompositional structure for Laurent series.

Abstract

A semidomain is a subsemiring of an integral domain. One can think of a semidomain as an integral domain in which additive inverses are no longer required. A semidomain $S$ is additively reduced if $0$ is the only invertible element of the monoid $(S,+)$, while $S$ is additively atomic if the monoid $(S,+)$ is atomic (i.e., every non-invertible element $s \in S$ can be written as the sum of finitely many irreducibles of $(S,+)$). In this paper, we describe the additively reduced and additively atomic semidomains $S$ for which every Laurent series $f \in S[[x^{\pm 1} ]]$ that is not a monomial can be written as the sum of at most three multiplicative irreducibles. In particular, we show that, for each $k \in \mathbb{N}$, every polynomial $f \in \mathbb{N}[x_1^{\pm 1}, \ldots, x_k^{\pm 1}]$ that is not a monomial can be written as the sum of two multiplicative irreducibles provided that $f(1, \ldots, 1) > 3$.

Theorems & Definitions (31)

• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5