Matrix Polynomial Factorization via Higman Linearization (2203.16978v1)

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 $\mathbb{F}q\langle x_1,x_2,\ldots,x_n \rangle$, 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 $\log_2q$ that computes a factorization of $M$ as a matrix product $M=M_1M_2\cdots M_r$, 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 $\mathbb{F}_q \langle x_1,x_2,\ldots,x_n \rangle$ 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.