Matrix Polynomial Factorization via Higman Linearization
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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× d matrix M whose entries M_ij are polynomials in the free noncommutative ring 𝔽_q⟨ x_1,x_2,…,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_2q that computes a factorization of M as a matrix product M=M_1M_2⋯ M_r, where each d× d matrix factor M_i is irreducible (in a well-defined sense) and the entries of each M_i are polynomials in 𝔽_q ⟨ x_1,x_2,…,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 𝔽[x]. When 𝔽 is a finite field the above result applies. When 𝔽 is the field of rationals we obtain a deterministic polynomial-time algorithm for the problem.
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