  For a positive real $\alpha$, we can consider the additive submonoid $M$ of
the 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, 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 $\alpha$ 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 and by Correa-Morris and Gotti
in 2022.
