Representation of $m$-gonal forms over $\mathbb N_0$ and a finiteness theorem for universal original version's $m$-gonal forms

In this article, we consider the representation of $m$-gonal forms over $\mathbb N_0$. We show that any $m$-gonal forms over $\mathbb N_0$ of rank $\ge 5$ is almost regular and ponder the sufficiently large integers which are indeed represented over $\mathbb N_0$ among the integers which are locally represented. And as its consequential result, we prove a finiteness theorem for universal (original polygonal number version's) $m$-gonal forms over $\mathbb N_0$.