Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq 4$ the ${\mathcal O}$-points on $V_k(A)$ are Zariski dense if $V_{k}(A)({\mathcal O})\neq\emptyset$ assuming the group of units ${\mathcal O}^{\times}$ is infinite. This shows that if $A$ can be written as a product of $k\geq 4$ elementary matrices, then this can be done in infinitely many ways in the strongest sense possible. This can then be combined with results on factoring into elementary matrices for ${\rm SL}_{2}({\mathcal O})$. One result is that for $k\geq 9$ the ${\mathcal O}$-points on $V_{k}(A)$ are Zariski dense if ${\mathcal O}^{\times}$ is infinite.

中文翻译：

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通过矩阵分解定义为初等矩阵的变体上的积分点

令 ${\mathcal O}$ 是数域 $K$ 中的 $S$-整数环。对于 $A\in\rm{SL}_{2}(\mathcal{O})$ 和 $k\geq 1$，我们在 ${\mathcal O}$ 上定义矩阵分解变体 $V_k(A)$将 $A$ 分解为 $k$ 基本矩阵的乘积参数化；定义 $V_k(A)$ 的方程是根据欧拉连续多项式编写的。我们表明 $V_k(A)$ 是具有感应纤维化结构的有理 $(k-3)$ 折叠。我们将这种几何结构与算术结果结合起来研究 $V_k(A)$ 的 ${\mathcal O}$ 点的 Zariski 闭包。我们证明对于 $k\geq 4$ $V_k(A)$ 上的 ${\mathcal O}$-点是 Zariski 稠密的，如果 $V_{k}(A)({\mathcal O})\neq\emptyset $ 假设单位组 ${\mathcal O}^{\times}$ 是无限的。这表明如果 $A$ 可以写成 $k\geq 4$ 初等矩阵的乘积，那么这可以在最强烈的意义上以无限多种方式完成。然后可以将其与分解为 ${\rm SL}_{2}({\mathcal O})$ 的初等矩阵的结果相结合。一个结果是，对于 $k\geq 9$，如果 ${\mathcal O}^{\times}$ 是无限的，则 $V_{k}(A)$ 上的 ${\mathcal O}$-点是 Zariski 稠密的。