  For every constant $d$, we design a subexponential time deterministic
algorithm that takes as input a multivariate polynomial $f$ given as a constant
depth algebraic circuit over the field of rational numbers, and outputs all
irreducible factors of $f$ of degree at most $d$ together with their respective
multiplicities. Moreover, if $f$ is a sparse polynomial, then the algorithm
runs in quasipolynomial time.
  Our results are based on a more fine-grained connection between polynomial
identity testing (PIT) and polynomial factorization in the context of constant
degree factors and rely on a clean connection between divisibility testing of
polynomials and PIT due to Forbes and on subexponential time deterministic PIT
algorithms for constant depth algebraic circuits from the recent work of
Limaye, Srinivasan and Tavenas.
