Stability and complexity of mixed discriminants

Abstract

We show that the mixed discriminant of $n$ positive semidefinite $n\times n$ real symmetric matrices can be approximated within a relative error $\epsilon >0$ in quasi-polynomial $n^{O(\ln n-\ln \epsilon )}$ time, provided the distance of each matrix to the identity matrix in the operator norm does not exceed some absolute constant ${\gamma }_0>0$. We deduce a similar result for the mixed discriminant of doubly stochastic $n$-tuples of matrices from the Marcus - Spielman - Srivastava bound on the roots of the mixed characteristic polynomial. Finally, we construct a quasi-polynomial algorithm for approximating the sum of $m$-th powers of principal minors of a matrix, provided the operator norm of the matrix is strictly less than 1. As is shown by Gurvits, for $m=2$ the problem is $\#P$-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.

Included an algorithm for computing the sum of powers of principal minors of a matrix