$D(N)$-quadruples in upper-triangular $2\times2$ integer matrices

Technical Summary: D(N)-Quadruples in Upper-Triangular 2 × 2 Integer Matrices

Problem Statement
The paper addresses the extension of Diophantine $D(n)$-$m$-tuples to the noncommutative ring $M_2(\mathbb{Z})$ of $2 \times 2$ integer matrices. In commutative rings, a $D(n)$-$m$-tuple is a set of distinct nonzero elements $\{a_1, \dots, a_m\}$ such that $a_i a_j + n$ is a perfect square for all $i < j$. A classical result in commutative rings establishes a strong link between the existence of $D(n)$-quadruples and the representability of $n$ as a difference of two squares ($n = x^2 - y^2$).

The authors investigate whether this relationship holds in the noncommutative setting of matrix rings. Specifically, they focus on the ring of upper-triangular integer matrices, $UT_2(\mathbb{Z})$, and explore two definitions of matrix $D(N)$-tuples:

1. Standard Product: Based on the ordered matrix product $A_i A_j + N = X_{ij}^2$ (distinguishing ascending and descending orders).
2. Jordan Product: Based on the symmetric Jordan product $A \circ B = \frac{1}{2}(AB + BA)$, defined as $A_i \circ A_j + N = X_{ij}^2$.

The primary focus is on Jordan $D(N)$-tuples within $UT_2(\mathbb{Z})$, where the square roots $X_{ij}$ must also belong to $UT_2(\mathbb{Z})$. The study distinguishes between degenerate tuples (where pairwise Jordan products are zero) and non-degenerate ones.

Methodology
The authors employ a combination of structural characterization, explicit construction, and modular obstruction techniques:

• Structural Characterization: They utilize a prior characterization (from reference [10]) of matrices in $UT_2(\mathbb{Z})$ representable as differences of two squares, which relies on congruence conditions modulo 16.
• Extension Formulas: They adapt the classical extension formula for triples. If $\{A, B\}$ is a Jordan $D(N)$-pair with $A \circ B + N = R^2$, they construct a triple $\{A, B, A+B \pm 2R\}$. This relies on the bilinearity of the Jordan product.
• Prototype Construction (Theorem 5): For matrices $N$ representable as $N = K^2 - A_0^2$, the authors construct a "prototype" quadruple of the form $\{A_0 + U_t, A_0 - U_t, 2A_0 + 2K, 5A_0 + 4K\}$, where $U_t$ is a nilpotent matrix. They prove that for infinitely many parameters $t$, this set forms a valid Jordan $D(N)$-quadruple.
• Modular Obstructions: For matrices $N$ not representable as differences of two squares, the authors analyze the scalar conditions on the diagonal entries ($a_i a_j + p = \square$ and $d_i d_j + q = \square$) and the integrality conditions on the upper-right entries. They use parity arguments and a specific lemma regarding vectors in $\mathbb{Z}_2^2$ (Lemma 7) to prove non-existence for specific congruence classes.
• Explicit Constructions: For certain congruence classes where non-existence cannot be proven, the authors provide explicit polynomial or integer constructions of quadruples, often utilizing scaling properties (Lemma 2) or adapting classical polynomial families (Section 5).

Key Contributions and Results

1. Existence for Difference-of-Squares Matrices:
   The paper proves that every matrix $N \in UT_2(\mathbb{Z})$ that can be represented as a difference of two squares in $UT_2(\mathbb{Z})$ admits infinitely many Jordan $D(N)$-quadruples (Theorem 5 and Corollary 6). This establishes the implication:
   $$N = K^2 - A^2 \implies \text{Existence of Jordan } D(N)\text{-quadruples}$$
   This is the first structure where this implication is established by combining a structural characterization of difference-of-squares representations with an explicit construction.

2. Non-Existence Results:
   The authors identify several infinite families of matrices $N \in UT_2(\mathbb{Z})$ that are not representable as differences of two squares and for which Jordan $D(N)$-quadruples do not exist. These non-existence results are derived via modular obstructions (Propositions 8, 9, 12, 13, 15, 17, 18).

   • Case I: $p \equiv 2 \pmod 4$ or $q \equiv 2 \pmod 4$.
   • Case II (Partial): $p, q$ odd, $p \equiv q \pmod 4$, $r$ odd, specifically when $p \equiv q \equiv 3 \pmod 4$.
   • Case III (Partial): $(p, q) \equiv (4, 4), (12, 12) \pmod{16}$ with specific conditions on $r$.
   • Case IV: $(p, q) \equiv (4, 12), (8, 8), (12, 4) \pmod{16}$ with $r$ odd.

3. Existence for Non-Difference-of-Squares Matrices:
   Crucially, the paper demonstrates that the "difference-of-squares" condition is not necessary for the existence of Jordan $D(N)$-quadruples in $UT_2(\mathbb{Z})$.

   • Proposition 10: Constructs a Jordan $D(N)$-quadruple for $N = \begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix}$ for any integer $m$. If $m$ is odd, $N$ is not a difference of two squares in $UT_2(\mathbb{Z})$, yet the quadruple exists.
   • Proposition 14: Proves that if the diagonal entries of $N$ are integer squares ($p=s^2, q=t^2$), then a Jordan $D(N)$-quadruple exists for every upper-right entry $r$, regardless of whether $N$ is a difference of two squares.
   • Proposition 16: Provides a family of quadruples for $N_t = \begin{pmatrix} 4 & 2t \\ 0 & 4 \end{pmatrix}$, which are representable as differences of squares only when $t$ is even, yet quadruples exist for all $t \neq 0$.

4. Polynomial Adaptations:
   Section 5 demonstrates how classical polynomial formulas for commutative $D(n)$-quadruples (specifically from Dujella [6]) can be adapted to the Jordan setting in $UT_2(\mathbb{Z})$ by solving linear Diophantine equations for the upper-right entries.

Significance and Claims
The paper claims to initiate the study of Diophantine $D(N)$-tuples in a genuinely noncommutative setting. Its primary significance lies in revealing that the relationship between difference-of-squares representations and the existence of $D(N)$-quadruples in matrix rings is more subtle than in the classical commutative setting.

While the implication "Difference of squares $\implies$ Existence of quadruples" holds for $UT_2(\mathbb{Z})$, the converse is false. The paper establishes that there are infinite families of matrices that are not differences of two squares yet still admit Jordan $D(N)$-quadruples. Conversely, there are matrices that are not differences of squares and for which no such quadruples exist.

The authors conclude that while the "rule" linking difference-of-squares representations to quadruple existence holds for a large class of matrices in $UT_2(\mathbb{Z})$, it is not a strict equivalence. The paper leaves certain congruence classes (specifically in Cases II and III where diagonal entries are not squares) as open problems, noting that both existence and non-existence phenomena occur within these residual classes.