Deterministic Depth-4 PIT and Normalization

In this paper, we initiate the study of deterministic PIT for ${\mathrm{\Sigma }}^{[k]}\mathrm{\Pi }\mathrm{\Sigma }{\mathrm{\Pi }}^{[\delta ]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for ${\mathrm{\Sigma }}^{[3]}\mathrm{\Pi }\mathrm{\Sigma }{\mathrm{\Pi }}^{[\delta ]}$ circuits, under the additional condition that one of the summands at the top $\mathrm{\Sigma }$ gate is squarefree. Our techniques are purely algebro-geometric: they do not rely on Sylvester–Gallai-type theorems, and our PIT result holds over arbitrary fields. The core of our proof is based on the normalization of algebraic varieties. Specifically, we carry out the analysis in the integral closure of a coordinate ring, which enjoys better algebraic properties than the original ring.