In an influential 1877 paper, Zolotarev asked and answered four questions about polynomial and rational approximation. We ask and answer two questions: what are the best rational approximants $r$ and $s$ to $\sqrt{z}$ and $\mbox{sign}(z)$ on the unit circle (excluding certain arcs near the discontinuities), with the property that $|r(z)|=|s(z)|=1$ for $|z|=1$? We show that the solutions to these problems are related to Zolotarev's third and fourth problems in a nontrivial manner.

中文翻译：

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佐洛塔列夫的第五和第六个问题

佐洛塔列夫（Zolotarev）在1877年的一篇颇具影响力的论文中提出并回答了关于多项式和有理逼近的四个问题。我们问和回答两个问题：单位圆上的最佳有理近似值$ r $和$ s $到$ \ sqrt {z} $和$ \ mbox {sign}（z）$是什么（不连续点附近的某些弧） ），具有$ | z | = 1 $的属性| $ | r（z）| = | s（z）| = 1 $？我们证明，这些问题的解决方案以非平凡的方式与佐洛塔列夫的第三个和第四个问题有关。