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 $UT_2(\mathbb Z)$.
Let $U$ be the ring of upper triangular $2\times2$ integer matrices. For $N\in U$, a Jordan $D(N)$-set is a set of distinct nonzero matrices in $U$ such that $(AB+BA)/2+N$ is a square in $U$ for any two distinct elements $A$ and $B$. We determine all $N$ for which an infinite Jordan $D(N)$-set exists. If no infinite se...
Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\te...
Let $R$ be non-commutative ring with unity. In this paper, we prove that a given matrix in $M_n(R)$ is a sum of $p$-th powers in $M_n(R)$, where $n<p$ if and only if its trace can be written as a sum of $p$-th powers in $M_n(R)$, for $n = 2,3,4$ and $p = 5$ and $n = 4,5,6$ for $p = 7$. This paper extends the results of...
A $m$-tuple with the property $D(n)$ is a tuple of $m$ positive integers $(a_1, a_2, \dots, a_m)$ such that $a_i a_j + n$ is an square, for $1 \le i<j \le m$. The $k$th triangular number is $T_k = \frac{k(k+1)}{2}$ for nonnegative integers $k$. We consider $D(1)$ tuples consisting only of triangular numbers. We prove t...
Let $K(n)$ be the generalized Kronecker quiver, and let $(T(n),\Omega)$ be the universal covering of $K(n)$, where $n\geq 3$ and $\Omega$ is a bipartite orientation. We introduce three different types of representations for quiver $(T(n),\Omega)$ by $K(n)$: maps are injective in a representation maps are all surjective...
If $M$ is a closed oriented manifold of even dimension, then its $n$th symmetric product $M(n)=M^n/S_n$ is a rational homology manifold for any $n \geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thes...
Julia Semikina, Don Zagier· 0 citations
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