Zero-sum Square Matrices

Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer m for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim supM(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k √ 2 2e exp(1/e) . Some nontrivial explicit values are also computed.