We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n -tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n -tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in R n to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A ``side benefit'' is a generalization of Rado's theorem on the existence of a linearly independent transversal.
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Gurvits, Samorodnitsky A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary. Discrete Comput Geom 27, 531–550 (2002). https://doi.org/10.1007/s00454-001-0083-2