论文标题
化粪池适当Zolotarev多项式的明确单变量和根治性参数化
An explicit univariate and radical parametrization of the septic proper Zolotarev polynomials in power form
论文作者
论文摘要
确定$ [-1,1] $上适当的$ n $ thgure Zolotarev多项式的明确单参数功率形式表示的问题可以追溯到P. L. Chebyshev。事实证明这很复杂，即使对于$ n $的少量值也是如此。 A. A. Markov（1889）以$ n = 2 $和$ n = 3 $知道了这样的代表。但是似乎已经以$ n = 4 $的方式了，似乎没有人真正相信可以找到明确的表格。实际上，正如A. Shadrin在2004年所说的那样，V. A. Markov在1892年。下一个较高的学位，$ n = 5 $和$ n = 6 $，直到最近才由G. Grasegger和N. Th解决。 VO（2017）分别由现代作者（2019）。在本文中，我们使用符号计算解决了$ n = 7 $的情况。 \ {2,3,4 \} $的学位的参数化是一个合理的，而对于$ n \ in \ {5,6,7 \} $，它是一个激进的。但是，从根本参数化中，$ n = 7 $需要特别注意，因为它不是一个简单的根本性参数。
The problem of determining an explicit one-parameter power form representation of the proper $n$-th degree Zolotarev polynomials on $[-1,1]$ can be traced back to P. L. Chebyshev. It turned out to be complicated, even for small values of $n$. Such a representation was known to A. A. Markov (1889) for $n=2$ and $n=3$. But already for $n=4$ it seems that nobody really believed that an explicit form can be found. As a matter of fact it was, by V. A. Markov in 1892, as A. Shadrin put it in 2004. The next higher degrees, $n=5$ and $n=6$, were resolved only recently, by G. Grasegger and N. Th. Vo (2017) respectively by the present authors (2019). In this paper we settle the case $n=7$ using symbolic computation. The parametrization for the degrees $n\in \{2,3,4\}$ is a rational one, whereas for $n\in \{5,6,7\}$ it is a radical one. However, the case $n=7$ among the radical parametrizations requires special attention, since it is not a simple radical one.
