Regular triangular forms of rank exceeding 3

TL;DR

This paper classifies all regular triangular forms with more than three variables, which are quadratic polynomials representing all locally representable positive integers.

Contribution

It provides a complete classification of regular triangular forms of rank exceeding 3, expanding understanding of quadratic forms in number theory.

Findings

Identified all regular triangular forms with more than three variables.

Established criteria for regularity in these forms.

Contributed to the theory of quadratic forms and their representability.

Abstract

A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we find all regular triangular forms of more than three variables.