Matrix Polynomial Factorization via Higman Linearization

Abstract

In continuation to our recent work on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results. (1) Given as input a full rank $d\times d$ matrix $M$ whose entries $M_{ij}$ are polynomials in the free noncommutative ring ${\mathbb{F}}_q⟨x_1,x_2,\dots ,x_n⟩$, where each $M_{ij}$ is given by a noncommutative arithmetic formula of size at most $s$, we give a randomized algorithm that runs in time polynomial in $d,s,n$ and ${\log }_{2}q$ that computes a factorization of $M$ as a matrix product $M=M_1{M}_2\cdots M_r$, where each $d\times d$ matrix factor $M_i$ is irreducible (in a well-defined sense) and the entries of each $M_i$ are polynomials in ${\mathbb{F}}_q⟨x_1,x_2,\dots ,x_n⟩$ that are output as algebraic branching programs. We also obtain a deterministic algorithm for the problem that runs in $poly(d,n,s,q)$. (2)A special case is the efficient factorization of matrices whose entries are univariate polynomials in $\mathbb{F}[x]$. When $\mathbb{F}$ is a finite field the above result applies. When $\mathbb{F}$ is the field of rationals we obtain a deterministic polynomial-time algorithm for the problem.