Regular triangular forms of rank exceeding 3

Abstract

A triangular form is an integer-valued quadratic polynomial of the form $a_1{P}_3\left(x_1\right)+a_2{P}_3\left(x_2\right)+\cdots +a_k{P}_3\left(x_k\right)$, where the coefficients $a_i$ are positive integers and $P_3\left(x\right)=x(x+1)/2$. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we determine all regular triangular forms of more than three variables.