The factorization of the permanent of a matrix with minimal rank in prime characteristic

It is known that any square matrix A of size n over a field of prime characteristic p that has rank less than n/(p − 1) has a permanent that is zero. We give a new proof involving the invariant X _p . There are always matrices of any larger rank with non-zero permanents. It is shown that when the rank of A is exactly n/(p − 1), its permanent may be factorized into two functions involving X _p .