Multivariate to bivariate reduction for noncommutative polynomial factorization

Abstract

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,

• 1.

  Given an n-variate noncommutative polynomial $f\in F〈X〉$ over a field $F$ as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial $g\in F〈x,y〉$; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.

  The reduction works both in the white-box and the black-box setting.

• 2.

  We show over the field of rationals that bivariate linear matrix factorization problem for $4\times 4$ matrices is at least as hard as factoring square-free integers and for $3\times 3$ matrices it is in polynomial time.