Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time

Abstract

Let {\mathcal B} be a linear space of matrices over a field {\mathbb spanned by n\times n 
matrices B_1, \dots, B_m. The non-commutative rank of {\mathcal B}$ is the minimum r\in {\mathbb N} such that there exists U\leq {\mathbb F}^n satisfying \dim(U)-\dim( {\mathcal B} (U))\geq 
n-r, where {\mathcal B}(U):={\mathrm span}(\cup_{i\in[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.