The rank of universal $m$-gonal forms

TL;DR

This paper investigates the minimal and maximal ranks of universal m-gonal forms for large m, providing key insights into their structure and classification.

Contribution

It determines the minimal rank of universal m-gonal forms and the maximal rank of proper universal m-gonal forms for sufficiently large m.

Findings

Identified the minimal rank of universal m-gonal forms.

Established the maximal rank of proper universal m-gonal forms.

Provided classifications for large m values.

Abstract

In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.