Permutation Polynomials: A Matrix Analogue of Schur′s Conjecture and a Survey of Recent Results

We consider a matrix analogue of Schur′s conjecture concerning permutation polynomials induced by polynomials with integral coefficients. For any fixed integer m ≥ 1 we consider polynomials with integral coefficients which induce permutations on the ring of all m × m matrices over the finite field Fp for infinitely many primes p. We also provide a survey of recent results concerning permutation polynomials over finite fields.