Based on a theorem of Bergman we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following: (1) In the white-box setting, given an n-variate noncommutative polynomial f in F<X> over a field F (either a finite field or the rationals) as an arithmetic circuit (or algebraic branching program), computing a complete factorization of f is deterministic polynomial-time reducible to white-box factorization of a noncommutative bivariate polynomial g in F<x,y>; the reduction transforms f into a circuit for g (resp. ABP for g), and given a complete factorization of g the reduction recovers a complete factorization of f in polynomial time. We also obtain a similar deterministic polynomial-time reduction in the black-box setting. (2) Additionally, we show over the field of rationals that bivariate linear matrix factorization of 4 x 4 matrices is at least as hard as factoring square-free integers. This indicates that reducing noncommutative polynomial factorization to linear matrix factorization (as done in our recent work [AJ22]) is unlikely to succeed over the field of rationals even in the bivariate case. In contrast, multivariate linear matrix factorization for 3 x 3 matrices over rationals is in polynomial time.

翻译：基于Bergman的一个定理，我们证明了多元非交换多项式因式分解可确定性地在多项式时间内归约到二元非交换多项式的因式分解。具体而言，我们证明了以下结论：（1）在白箱设定下，给定域F（有限域或有理数域）上由算术电路（或代数分支程序）表示的n元非交换多项式f ∈ F<X>，计算f的完全分解可确定性地在多项式时间内归约到非交换二元多项式g ∈ F<x,y>的白箱因式分解；该归约将f转化为g的电路（或g的ABP），并且给定g的完全分解，归约可在多项式时间内恢复出f的完全分解。我们在黑箱设定下也得到了类似的确定性多项式时间归约。（2）此外，我们证明了在有理数域上，4×4矩阵的二元线性矩阵分解的难度不低于分解无平方因子整数。这表明将非交换多项式因式分解归约到线性矩阵分解（如我们近期工作[AJ22]所做）在有理数域上即使对于二元情形也难以成功。相比之下，有理数域上3×3矩阵的多元线性矩阵分解可在多项式时间内完成。