On permutation polynomials in two variables over \(\mathbb Z /{p^2}\mathbb Z\)
Polynomials \(F_1,\dots, F_k\) in \(n\) variables over a finite commutative ring are said to form an orthogonal system if all pre-images of the map from \(\mathbb R^n\) to \(\mathbb R^k\) induced by \(F_1,\dots, F_k\) have the same size. An orthogonal system \(F_1,\dots, F_k\) for which there exist polynomials \(F_{k+1},\dots, F_n\) such that \(F_1,\dots, F_n\) induces a permutation of \(\mathbb R^n\) is called a strong orthogonal system. In particular, a single polynomial \(F\) is called a (strong) permutation polynomial (abbr. PP) if it forms a (strong) orthogonal system. In [Finite Fields Appl. 13, No. 1, 113--120 (2007; Zbl 1109.11058)], the authors gave a condition for PP over local rings to be strong, which generalized \textit{W. Nöbauer}'s theorem [Monatsh. Math. 69, 230--238 (1965; Zbl 0131.04201)]. In this article the authors analyse polynomials in two variables over \(R = \mathbb Z/p^2\mathbb Z\). The results in detail are quite sophisticated, in nature the authors show which polynomials in \(\mathbb Z/p\mathbb Z[X_1,X_2]\) are reductions of some PP's in \(\mathbb Z/p^2\mathbb Z[X_1,X_2]\), and for such a polynomial \(F \in \mathbb Z/p\mathbb Z[X_1,X_2]\) which liftings are PP's in \(\mathbb Z/p^2\mathbb Z[X_1,X_2]\), and they characterize the orbits of the set of PP's in 2 variables over \(R = \mathbb Z/p^2\mathbb Z\) under the group of the permutations of \(R^2\).
scientific article; zbMATH DE number 5601615
scientific article