Given a basis for a linear subspace U of nxn matrices, we study the problem of either producing a rank-one matrix in U, or else certifying that none exist. While this problem is NP-Hard in the worst case, we present a polynomial time algorithm to solve this problem in the generic setting under a mild condition on the dimension of U. Our algorithm is based on Hilbert’s Nullstellensatz and a “lifted” adaptation of the simultaneous diagonalization algorithm for tensor decompositions. We extend our results to the more general setting in which the set of rank-one matrices is replaced by an algebraic set. Time permitting, we will discuss applications to quantum separability testing and tensor decompositions. This talk is based on joint work with Harm Derksen, Nathaniel Johnston, and Aravindan Vijayaraghavan.
