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On the Monotone Column Permanent conjecture

Abstract : We discuss some recent progress on the Monotone Column Permanent (MCP) conjecture. We use a general method for proving that a univariate polynomial has real roots only, namely by showing that a corresponding multivariate polynomial is stable. Recent connections between stability of polynomials and the strong Rayleigh property revealed by Brändén allows for a computationally feasible check of stability for multi-affine polynomials. Using this method we obtain a simpler proof for the $n=3$ case of the MCP conjecture, and a new proof for the $n=4$ case. We also show a multivariate version of the stability of Eulerian polynomials for $n \leq 5$ which arises as a special case of the multivariate MCP conjecture.
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James Haglund, Mirkó Visontai. On the Monotone Column Permanent conjecture. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.443-454. ⟨hal-01185436⟩

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