math.CO — Combinatorics

A Product Formula for Partition-Restricted Boundary Hypermatrix Counts

Contributed by YURAN XIU

We prove the Koprowski--Lewis product conjecture for nondegenerate $2\times(k+1)\times k$ hypermatrices over finite fields with two partition-shaped layers of prescribed zeros. The result holds for every positive $k$, every finite field, and every admissible pair of partitions. After fixing the first slice, nondegeneracy is equivalent to simplicity of a partial linear map. Applying the simple-extension theorem of Arora, Ram, and Venkateswarlu inside the successive allowed row spaces gives the product directly. We include an elementary proof of that extension theorem and justify the equivalence with the defining projective critical locus of the hyperdeterminant. The complete counting statement and its semantic equivalences have been checked in Lean. HAL deposit: hal-05800194v1.