AI 中文总结
该研究将Gustafson等人关于矩阵分解为对合乘积的经典结果推广到连续或全纯函数环上的特殊线性群，证明零同伦是有限对合分解的充要条件，并给出依赖于矩阵大小和参数空间维度的上界。
AI 中文摘要
Gustafson、Halmos和Radjavi在1976年的一个著名结果表明，$SL_n(\mathbb{C})$中的任何矩阵$A$至多是4个对合的乘积。我们考虑这一结果在Vaserstein和Gromov精神下的自然连续且全纯的参数依赖。我们的主要结果表明，在连续或全纯函数环上的特殊线性群中，零同伦恰好刻画了那些可以表示为有限个对合乘积的矩阵。我们给出了一个上界$I(n, d)$，该上界仅依赖于矩阵的大小$n$和参数空间的维度$d$，用于分解所需的对合数量。我们的上界并非最优，但我们证明对于固定的$n$，当$d\to \infty$时，最优上界趋于无穷。在某些情况下，我们能够确定最优界。
英文摘要
It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size $n$ of the matrices and the dimension $d$ of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed $n$ the optimal upper bound tends to infinity as $d\to \infty$. In some cases we are able to determine the optimal bound.
