菱形行列式与Somos序列
Diamond Determinants and Somos Sequences
AI总结：
本文通过初等方法证明了6阶Somos序列的有限秩性质，并发现了7阶Somos序列的新有限秩性质，同时提出了对高阶Gale-Robinson序列的推广猜想。
摘要：
一个$n$阶Somos序列由宽度为$n+1$的二次递推定义。这些序列在小$n$时的某些显著性质与由它们构成的特定矩阵具有有限秩有关。我们给出了6阶有限秩性质的初等证明，此前仅借助超椭圆函数理论的高级工具才能建立。我们的方法还得到了7阶Somos序列的一个新的有限秩性质。此外，我们猜想这些结果可以推广到更高阶的Gale-Robinson子类序列。
英文摘要：
A Somos sequence of order $n$ is defined by a quadratic recurrence of width $n + 1$. Some of the remarkable properties of these sequences for small $n$ are tied to certain matrices built out of them being of finite rank. We give an elementary proof of the finite-rank property for order $6$, previously only established with the help of advanced machinery from the theory of hyperelliptic functions. Our method also yields a new finite-rank property for the Somos sequences of order $7$. In addition, we conjecture generalisations of these results to higher orders, for the subclass of Gale-Robinson sequences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
