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^±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 N, every polynomial f in N[x₁^±1, ..., x_k^±1] that is not a monomial can be written as the sum of two multiplicative irreducibles provided that f(1, ..., 1) > 3.

The authors thank the anonymous referee whose suggestions helped improve this paper. The first author was supported by NSF grant DMS 2154223. The second author was supported by a University of California President's Postdoctoral Fellowship.