Towards Deterministic Algorithms for Constant-Depth Factors of Constant-Depth Circuits

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.