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

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