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Friday, February 18, 2011 |
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Proof of the monotone column permanent conjecture |
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Let $A$ be a square matrix of real numbers which are weakly decreasing down each column, let $J$ be the all-ones matrix of the same size, and let $z$ be an indeterminate. In 1999, Jim Haglund, Ken Ono, and I conjectured that the permanent of the matrix $zJ + A$ is a polynomial with only real roots. Last year, Petter Brändén, Jim, Mirkó Visontai and I proved a multivariate version of this. This is a relatively simple application (suitable as an introduction) of one of my favourite subjects -- the theory of multivariate stable polynomials. |