On strong orthogonal systems and weak permutation polynomials over finite commutative rings

We study two kinds of orthogonal systems of polynomials over finite commutative rings and get two fundamental results. Firstly, we obtain a necessary and sufficient condition for a system of polynomials (over a fixed finite commutative ring R) to form a strong orthogonal system. Secondly, for a pair (R,n) of a finite local ring R and an integer n>1, we get an easy criterion to check whether every weak permutation polynomial in n variables over R is strong.