On permutation polynomials in two variables over \(\mathbb Z /{p^2}\mathbb Z\)
type: Item
instance of: scholarly article
title: On permutation polynomials in two variables over \(\mathbb Z /{p^2}\mathbb Z\)
zbMATH Open document ID: 1259.11111
author: Qijiao Wei
author: Qifan Zhang
publication date: 2009-09-03T00:00:00Z
review text: 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 \(R^2\).
reviewed by: Wilfried Meidl
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 5601615
zbMATH Keywords: permutation polynomial
zbMATH Keywords: Witt polynomial
zbMATH Keywords: Witt vector ring
MaRDI profile type: Publication
full work available at URL: s10114-009-6526-z
OpenAlex ID: W2015096048
cites work: Q3216750
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cites work: On strong orthogonal systems and weak permutation polynomials over finite commutative rings
cites work: When are weak permutation polynomials strong?
cites work: Polynomial functions and permutation polynomials over some finite commutative rings
cites work: Q3856855
DOI: 10.1007/S10114-009-6526-Z
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5601615