Factorizations of polynomials with integral non-negative coefficients

TL;DR

This study analyzes the monoid $\mathbb{N}_0[x]^*$ of polynomials with non-negative integer coefficients, showing it is not half-factorial and not Krull, yet exhibits a Krull-like structure via derivations and connections to the Weyl algebra $A_1(\mathbb{Z})$. It identifies the prime elements as the positive integers and $x$, and provides a detailed decomposition $\mathbb{N}_0[x]^*\cong\mathbb{N}\times P$ with $P\cong\mathbb{N}\times P_0$, along with a symmetry $\varphi$; the paper also develops an ideal/prime-ideals framework, including chains of principal ideals and valuations, and presents a constructive, base-$a$-representation algorithm for factorizations. By contrasts with Krull theory, it demonstrates non-existence of ACC on ideals and non-root-closedness, yet offers a derivation-based paradigm that parallels Krull decompositions. The results yield structural insights and a practical algorithm for factoring within $\mathbb{N}_0[x]^*$, linking factorization theory to differential-operator techniques.

Abstract

We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N_0$ with derivations into $\mathbb N_0$. We study ideals, chain of ideals, prime ideals and prime elements of $\mathbb N_0[x]^*$. Our monoid $\mathbb N_0[x]^*$ is a submonoid of the multiplicative monoid of the ring $\mathbb Z[x]$, which is a left module over the Weyl algebra $A_1(\mathbb Z)$.

Theorems & Definitions (23)

• Remark 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 3.1
• proof