Year: 2026
Type: preprint
Abstract: We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\mathbb Z)$ of $2\times2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\circ B=\frac12(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $UT_2(\mathbb Z)$, 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 $UT_2(\mathbb Z)$.
Source: arXiv (Cornell University)
Authors: Dujella, Andrej, Franušić, Zrinka
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204890874
Cites:
Cited by:
Related to: