Factorizations of polynomials with integral non-negative coefficients

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 $A_1(\mathbb Z)$.