Abstract
Let A be an n-by-n matrix of real numbers which are weakly decreasing down each column, Z n = diag(z 1,…, z n ) a diagonal matrix of indeterminates, and J n the n-by-n matrix of all ones. Weprove that per(J n Z n +A) is stable in the z i , resolving a recent conjecture of Haglund and Visontai. This immediately implies that per(zJ n ) is a polynomial in z with only real roots, an open conjecture of Haglund, Ono, and Wagner from 1999. Other applications include stability of a multivariate Eulerian polynomial, a new proof of Grace’ apolarity theorem, and new permanental inequalities.
Mathematics Subject Classification (2000).15A15; 32A60, 05A05, 05A15.
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Brändén, P., Haglund, J., Visontai, M., Wagner, D.G. (2011). Proof of the Monotone Column Permanent Conjecture. In: Brändén, P., Passare, M., Putinar, M. (eds) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0142-3_5
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DOI: https://doi.org/10.1007/978-3-0348-0142-3_5
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