We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of $f$ computable by constant-depth circuits. This list $L$ might also include circuits that are spurious: they either do not correspond to factors of $f$ or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of $f$ and a factor $g$, which serves as a proxy for the resultant of $g$ and $f/g$, with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of $f$ and $g$. This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas, helps us derandomize one key step of multivariate polynomial factorization algorithms - that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.

翻译：我们设计了一个确定性次指数时间算法，该算法输入一个由有理数上的常深度电路计算的多变量多项式 $f$，并输出一个电路列表 $L$（深度无界且可能包含除法门），其中包含所有可由常深度电路计算的 $f$ 的不可约因子。列表 $L$ 也可能包含虚假电路：它们要么不对应于 $f$ 的因子，要么甚至未良好定义，例如除法门的输入是计算恒为零多项式的子电路。我们算法的关键技术要素是 $f$ 与因子 $g$ 的伪结式概念，它作为 $g$ 与 $f/g$ 的结式的一种代理，其优势在于伪结式的电路复杂度与 $f$ 和 $g$ 的电路复杂度相当。这一概念可能具有独立意义，结合 Limaye、Srinivasan 和 Tavenas 的最新成果，它帮助我们实现了多变量多项式分解算法中一个关键步骤的去随机化——即在输入多项式及其感兴趣的不可约因子均具有小常深度电路的情况下，确定性寻找牛顿迭代的合适起始点。