On the additive structure of algebraic valuations of polynomial semirings

TL;DR

This paper investigates the additive structure and factorization properties of algebraic valuations in polynomial semirings, providing criteria for atomicity, finite and bounded factorizations, and unique factorization, with characterizations involving minimal polynomials.

Contribution

It offers a comprehensive analysis of when polynomial valuation monoids are atomic, finitely factorable, or uniquely factorable, including explicit descriptions and conditions related to minimal polynomials.

Findings

$ N_0[eta] $ is atomic under certain conditions.

Equivalence of BFM and FFM properties with ACCP in $ 0[eta]$.

Characterizations of UFM involving minimal polynomials.

Abstract

In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[\alpha]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $\mathbb{N}_0[\alpha]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $\mathbb{N}_0[\alpha]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $\mathbb{N}_0[\alpha]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $\alpha$. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.