论文信息 - The Van der Waerden conjecture for mixed discriminants

The Van der Waerden conjecture for mixed discriminants

We prove that the mixed discriminant of doubly stochastic n-tuples of semidefinite hermitian n×n matrices is bounded below by n!nn and that this bound is uniquely attained at the n-tuple (1nI,…,1nI). This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.

Leonid Gurvits

[1] L. Lovász,et al. Geometric Algorithms and Combinatorial Optimization , 1981 .

[2] D. Falikman. Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix , 1981 .

[3] Alex Samorodnitsky,et al. A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary , 2002, Discret. Comput. Geom..

[4] Alex Samorodnitsky,et al. A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume , 2000, STOC '00.

[5] Leonid Gurvits. Classical deterministic complexity of Edmonds' Problem and quantum entanglement , 2003, STOC '03.

[6] Ravindra B. Bapat,et al. Mixed discriminants of positive semidefinite matrices , 1989 .

[7] R. Schneider. Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[8] L. Gårding. An Inequality for Hyperbolic Polynomials , 1959 .

[9] Shmuel Friedland,et al. A lower bound for the permanent of a doubly stochastic matrix , 1979 .

[10] G. Egorychev. The solution of van der Waerden's problem for permanents , 1981 .

[11] Leonid Gurvits. Combinatorial and algorithmic aspects of hyperbolic polynomials , 2004, Electron. Colloquium Comput. Complex..

[12] David London,et al. Some notes on the van der Waerden conjecture , 1971 .