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.

翻译：对于每个常数 $d$，我们设计了一个亚指数时间确定性算法，该算法输入一个以有理数域上常数深度代数电路形式给出的多元多项式 $f$，并输出 $f$ 的所有次数不超过 $d$ 的不可约因式及其各自的重数。此外，若 $f$ 为稀疏多项式，则算法运行时间为拟多项式时间。我们的结果基于多项式恒等测试（PIT）与多项式因式分解在常数次因式背景下更精细的联系，并依赖于 Forbes 提出的多项式整除性测试与 PIT 之间的清晰联系，以及 Limaye、Srinivasan 和 Tavenas 近期工作中关于常数深度代数电路的亚指数时间确定性 PIT 算法。