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

TL;DR

This paper studies the representation of $m$-gonal forms over non-negative integers, proving a finiteness theorem for universal forms and showing that forms of rank at least five are almost regular, with large integers being represented under certain conditions.

Contribution

It establishes a finiteness theorem for universal $m$-gonal forms over $ _0$ and demonstrates that forms of rank ≥ 5 are almost regular.

Findings

Forms of rank ≥ 5 are almost regular.

Large integers locally represented are also globally represented.

Finiteness theorem for universal $m$-gonal forms.

Abstract

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 $\mathbb N_0$.