Factorization in additive monoids of evaluation polynomial semirings

For a positive real α, we can consider the additive submonoid M of line that is generated by nonnegative powers α. When α transcendental, unique factorization monoid. However, when algebraic, may not be atomic, and even it contain elements having more than one (i.e., decomposition as sum irreducibles). The main purpose this paper to study phenomenon multiple factorizations inside M. algebraic but rational, arithmetic in highly interesting complex. In order arrive conclusion, investigate various invariants M, including sets lengths, Betti elements, catenary degrees. Our investigation gives continuity recent studies carried out Chapman et al. 2020 Correa-Morris Gotti 2022.