The question of which elements of the open unit disc occur as eigenvalues of n-by-n doubly stochastic matrices has long been open, in spite of a number of intriguing partial results. By enhancing a natural, but slightly false, conjecture, we gain some new computational insights into the problem. We also apply the classical field of values to give some partial results on both the necessity of certain component polygons of $P{M}_n$ for other cycle lengths and the accuracy of the Perfect-Mirsky Conjecture in the neighborhoods of corners.

中文翻译：

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双随机单特征值问题的新观点

尽管有许多有趣的部分结果，但开放单位圆盘的哪些元素作为n × n双随机矩阵的特征值出现的问题长期以来一直悬而未决。通过增强自然但略微错误的猜想，我们获得了对该问题的一些新的计算见解。我们还应用经典的值域来给出一些关于某些组件多边形的必要性的部分结果$P{米}_n$对于其他循环长度和完美米尔斯基猜想在角落附近的准确性。