$D(N)$-quadruples in upper-triangular $2\times 2$ integer matrices

Abstract

We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\mathbb{Z})$ of $2\times 2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\circ B=\frac{1}{2}(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $U{T}_2(\mathbb{Z})$, where squares admit a particularly simple description. Motivated by the classical connection between representations of $n$ as a difference of two squares and the existence of $D(n)$-quadruples in commutative rings, we investigate the existence of Jordan $D(N)$-quadruples in $U{T}_2(\mathbb{Z})$.