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There was a long standing conjecture stating that the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of regular $k$-gons anchored at $1$ in the unit disc for $2 \leq k \leq n$.

Mashreghi and Rivard showed that this conjecture is wrong for $n = 5$, cf. Linear and Multilinear Algebra, Volume 55, Number 5, September 2007 , pp. 491-498.

Have we made progress since then, beyond $n=5$, or for $n=4$? ($n=2,3$ is pretty simple).

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I don't know. Math Reviews shows only one paper citing Mashreghi & Rivard, and judging by the review that paper doesn't go in the direction you ask about. If after a while you get no answer here, I think your question would be appropriate for MathOverflow (being careful to notify both sites of the double-posting). –  Gerry Myerson Nov 9 '11 at 3:11
Thanks Gerry. I'm not very familiar with what's more appropriate for MathOverflow vs. MathStackExchange, but I'll follow your advice. –  Nathan Portland Nov 9 '11 at 3:29
Your question is perfectly appropriate here, and you may well get a superb answer here - I certainly hope so. I'm just saying if nothing materializes here in a couple of days.... –  Gerry Myerson Nov 9 '11 at 3:48

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