Factorization in Additive Monoids of Evaluation Polynomial Semirings

TL;DR

This paper explores the complex factorization properties of additive monoids generated by powers of algebraic numbers, revealing phenomena like non-uniqueness of factorizations and analyzing various invariants.

Contribution

It provides a detailed study of factorization behavior in monoids generated by algebraic numbers, extending recent research and highlighting complex non-unique factorizations.

Findings

Algebraic but not rational $eta$-generated monoids exhibit complex factorization behavior.

Investigation of invariants like sets of lengths, Betti elements, and catenary degrees.

Continuity with recent studies by Chapman et al. and Correa-Morris and Gotti.

Abstract

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.