Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time

Let {mathcal B} be a linear space of matrices over a field {mathbb spanned by ntimes n matrices B_1, dots, B_m. The non-commutative rank of {mathcal B}$ is the minimum rin {mathbb N} such that there exists Uleq {mathbb F}^n satisfying dim(U)-dim( {mathcal B} (U))geq n-r, where {mathcal B}(U):={mathrm span}(cup_{iin[m]} B_i(U)). Computing the non-commutative rank generalizes some well-known problems including the bipartite graph maximum matching problem and the linear matroid intersection problem. In this paper we give a deterministic polynomial-time algorithm to compute the non-commutative rank over any field {mathbb F}. Prior to our work, such an algorithm was only known over the rational number field {mathbb Q}, a result due to Garg et al, [GGOW]. Our algorithm is constructive and produces a witness certifying the non-commutative rank, a feature that is missing in the algorithm from [GGOW]. Our result is built on techniques which we developed in a previous paper [IQS1], with a new reduction procedure that helps to keep the blow-up parameter small. There are two ways to realize this reduction. The first involves constructivizing a key result of Derksen and Makam [DM2] which they developed in order to prove that the null cone of matrix semi-invariants is cut out by generators whose degree is polynomial in the size of the matrices involved. We also give a second, simpler method to achieve this. This gives another proof of the polynomial upper bound on the degree of the generators cutting out the null cone of matrix semi-invariants. Both the invariant-theoretic result and the algorithmic result rely crucially on the regularity lemma proved in [IQS1]. In this paper we improve on the constructive version of the regularity lemma from [IQS1] by removing a technical coprime condition that was assumed there