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 $\nu$th powers of principal minors of a matrix, for an integer $\nu \geq 1$, provided the operator norm of the matrix is strictly less than 1. As is shown by Gurvits, for $\nu =2$ the problem is $\#P$-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.