New views of the doubly stochastic single eigenvalue problem

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 (Formula presented.) for other cycle lengths and the accuracy of the Perfect-Mirsky Conjecture in the neighborhoods of corners.