Enumerating local permutation polynomials over residue class rings
type: Item
instance of: scholarly article
title: Enumerating local permutation polynomials over residue class rings
zbMATH Open document ID: 1190.13019
author: Jiangmin Pan
author: Cai Heng Li
published in: Taiwanese Journal of Mathematics
publication date: 2009-11-30T00:00:00Z
review text: For a prime \(p\) and a positive integer \(n\), let \(f(x,y)\) denote a polynomial in two variables over \(\mathbb{Z}/p^n \mathbb{Z}\). Then \(f\) is called a local permutation polynomial (LPP) modulo \(p^n\) if for each \(a\) in \(\mathbb{Z}/p^n \mathbb{Z}\) the polynomials (in one variable) \(f(a,y)\) and \(f(x,a)\) are permutation polynomials, i.e., the respective induced maps from \(\mathbb{Z}/p^n \mathbb{Z}\) to itself are bijective. The number of LPPs and the degree of LPPs are investigated. We sketch one of the two main results of the paper (the other one deals with the degree). For a fixed prime \(p\), let \(\ell_n\) denote the number of LPPs modulo \(p^n\) (identifying those giving rise to the same map). Then \(\ell_2 = p^{p^2}(p-1)^{2p^2}\ell_1\) and for \(n>2\) one has \(\ell_n = p^{\omega_n} \ell_{n-1}\) with \(\omega_n\) some constant whose definition is too long to be recalled here.
Mathematics Subject Classification ID: 13M10
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 5640589
zbMATH Keywords: permutation polynomial
zbMATH Keywords: local permutation polynomial
reviewed by: Wolfgang A. Schmid
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500405545
OpenAlex ID: W4238357563
DOI: 10.11650/TWJM/1500405545
description: scientific article; zbMATH DE number 5640589