On strong orthogonal systems and weak permutation polynomials over finite commutative rings (Q865385)

Let \(f_1, \dots, f_k\) be \(k\) polynomials in \(n\) variables over a finite commutative ring \(R\). If they induce a uniform map from \(R^n\) to \(R^k\) then they are said to form a weak orthogonal system over \(R\) and they are said to form a strong orthogonal system over \(R\) if there additionally exist polynomials \(f_{k+1},\dots, f_n\) such that \(f_1,\dots,f_n\) induce a permutation of \(R^n\). If \(k = 1\) then the polynomial in \(n\) variables is called a weak (strong) permutation polynomial. As main result the authors prove that \(k\) polynomials \(f_1, \dots, f_k\) in \(n\) variables over a finite commutative local ring \(R\) with maximal ideal \(M\) generated by \(r\) elements (where \(r\) is chosen minimal) form a strong orthogonal system over \(R\) if and only if \(f_1\bmod M, \dots, f_k\bmod M\) form a weak orthogonal system over \(R/M\) and the Jacobi matrix (\(f_1^\prime(x)\bmod M, \dots, f_k^\prime(x)\bmod M\)) has rank \(k\) everywhere. Furthermore if \(n \leq r\), then every weak permutation polynomial in \(R[X_1,\dots,X_n]\) is strong.