We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}. \] This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of $r$ terms, where each term is a $d$-th power of an $s$-sparse polynomial of degree $δ$. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when $d>r^2$ and yields explicit hitting sets of size $O(r^4 s^4 n^2 d δ^3)$. The reconstruction algorithm runs in time $\textrm{poly}(n,s,d)$ under the condition $d=Ω(r^4δ)$, and in particular it tolerates polynomially large top fan-in $r$ and bottom degree $δ$. Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-$4$ powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set.

翻译：本文研究深度为$4$的算术电路确定性多项式恒等式检验（PIT）与重构算法，该电路形式为\[ Σ^{[r]}\!\wedge^{[d]}\!Σ^{[s]}\!Π^{[δ]}。\]该模型推广了华林分解与对角电路，并刻画了低度稀疏多项式幂次和的结构。具体而言，每个电路计算$r$项之和，其中每一项均为度为$δ$的$s$-稀疏多项式的$d$次幂。此模型亦涵盖张量分解与基于矩的学习任务（如混合模型与子空间学习）中出现的代数表示。我们针对该模型给出了最坏情况下的确定性PIT与重构算法。当$d>r^2$时，我们的PIT构造产生大小为$O(r^4 s^4 n^2 d δ^3)$的显式命中集。重构算法在$d=Ω(r^4δ)$条件下以$\textrm{poly}(n,s,d)$时间运行，特别地，该算法可容忍多项式级大的顶层扇入$r$与底层度$δ$。上述结果在特征为零或充分大特征的域上均成立。这些算法首次为具有无界顶层扇入的深度-$4$幂次电路提供了多项式时间确定性解。特别地，重构结果改进了先前需要非退化或平均情况假设的工作。PIT构造基于函数域ABC定理（Mason-Stothers定理），该定理保证了稀疏多项式高次幂在适当投影后的线性无关性。重构算法将此定理与基于Wronskian的微分算子、其核的结构性质以及Klivans-Spielman命中集的鲁棒版本相结合。