论文标题
有效地界限某些$ 2 \ times 2 $单数矩阵，由真实二次数字戒指上的单数矩阵
Effectively bounded idempotent generation of certain $2 \times 2$ singular matrices by idempotent matrices over real quadratic number rings
论文作者
论文摘要
令$ k = \ mathbb {q}（\sqrtα）$是一个真正的二次数字字段，其中$α$是一个不含正方形的正整数。令$ \ Mathcal {o} _k $为$ k $的整数环。在本文中，我们证明了一套$ 2 \ times 2 $单矩阵，其中包含$ \ MATHCAL {O} _K $的条目可以写入有限数量的iDempotent矩阵的产物。我们的主要定理可以看作是Cossu和Zanardo对最新结果的概括，该结果研究了通过$ \ Mathcal {O} _K $的有限生成某些奇异矩阵，而不是通过$ \ nathcal calper {o} $ as as os phatsion fived of $ \ mathcal {o} _k $ ass iDempotent odempotent Indempotent Indempotent Generation {
Let $k = \mathbb{Q}(\sqrtα)$ be a real quadratic number field, where $α$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ instead of bounded generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ as considered in this paper.
