  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.
