Matrix Polynomial Factorization via Higman Linearization

TL;DR

This paper presents randomized and deterministic algorithms for factorizing matrices of noncommutative polynomials, with applications to matrices over finite fields and the rationals, advancing noncommutative polynomial factorization methods.

Contribution

It introduces efficient algorithms for factorizing matrices of noncommutative polynomials, including a randomized polynomial-time method and a deterministic approach for specific cases.

Findings

Randomized algorithm runs in polynomial time in matrix size, formula size, number of variables, and log of field size.

Deterministic algorithm exists for matrices over finite fields and rationals, with polynomial complexity.

Special case: efficient factorization of matrices with univariate polynomial entries.

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.