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For a positive real $\alpha$, we can consider the additive submonoid $M$ ofthe real line that is generated by the nonnegative powers of $\alpha$. When$\alpha$ is transcendental, $M$ is a unique factorization monoid. However, when$\alpha$ is algebraic, $M$ may not be atomic, and even when $M$ is atomic, itmay contain elements having more than one factorization (i.e., decomposition asa sum of irreducibles). The main purpose of this paper is to study thephenomenon of multiple factorizations inside $M$. When $\alpha$ is algebraicbut not rational, the arithmetic of factorizations in $M$ is highly interestingand complex. In order to arrive to that conclusion, we investigate variousfactorization invariants of $M$, including the sets of lengths, sets of Bettielements, and catenary degrees. Our investigation gives continuity to recentstudies carried out by Chapman, et al. in 2020 and by Correa-Morris and Gottiin 2022.

Factorization in Additive Monoids of Evaluation Polynomial Semirings