A Goldbach theorem for Laurent series semidomains

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 $f\in \mathbb{N}[x_1^{\pm 1},\dots ,x_k^{\pm 1}]$ that is not a monomial can be written as the sum of two multiplicative irreducibles provided that $f(1,\dots ,1)>3$.

arXiv comment17 pages