Zero-sum square matrices (Q1864564)

Let \(A\) be a matrix over the integers, and let \(p\) be a positive integer. A submatrix \(B\) of \(A\) is zero-sum \(\text{mod }p\) if the sum of earch 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 \(\text{mod }p\). In this paper the authors determine several upper and lower bounds for \(M(p,k)\). They show that \(M(2,k)\) is linear in \(k\). They also show that \(M(3,k)\) is linear in \(k\) for infinitely many values of \(k\). In particular, they prove that \(\limsup M(2,k)/k\leq 4\), \(\liminf M(3,k)/k\leq 20\), and that \(M(p,k)\geq {k\sqrt{2}\over 2e} \exp(1/e)^{p/2}\). Some nontrivial explicit values are also computed.