Linearized polynomial maps over finite fields
type: Item
instance of: scholarly article
title: Linearized polynomial maps over finite fields
zbMATH Open document ID: 1305.14029
author: Joost Berson
published in: Journal of Algebra
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.1137
review text: Let \(F\) be a finite field. A polynomial \(P\in F[X_1,\dots,X_n]\) s called linearized if the induced map \(K^n\longrightarrow K\) is \(F\)-linear for every extension \(K/F\). In this paper, the author studies the properties of linearized polynomials and maps \(K^n\longrightarrow K^n\) defined by an \(n\)-tuple of such polynomials (linearized maps). He starts with the observation that the set of all linearized polynomials is linearly generated by monomials \(X_i^{p^m}\) (\(i=1,2,\dots,n; m=0,1,\dots\)) (Proposition 3.2), and shows that linearized maps are in a one-to-one correspondence with matrices over \(F[X]\) (Remark 3.5). It is shown moreover that an analogue of the Jacobian Conjecture can be established for linearized maps (Corollary 4.1), and all such invertible maps are tame (Theorem 4.3). Corollary 4.5 shows that every linearized polynomial is a coordinate of an invertible map, in Corollary 4.10 it is proved that if \(I\) is an ideal of \(F[X]\) generated by linearized polynomials, then \(F[X]/I\) is a finitely generated \(F\)-algebra if and only if is is a domain, and in Corollary 4.18 an analogue of the Linearization Conjecture is established for linearized maps. In the final section the author considers polynomials without mixed terms and maps defined by them, and shows the truth of the Jacobian Conjecture for such maps.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6329100
zbMATH Keywords: affine space
zbMATH Keywords: polynomials over finite fields
zbMATH Keywords: linearized polynomial
zbMATH Keywords: group of polynomial automorphisms
zbMATH Keywords: group of tame automorphisms
MaRDI profile type: Publication
OpenAlex ID: W2017652104
arXiv ID: 1201.1137
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DOI: 10.1016/J.JALGEBRA.2013.10.013
description: scientific article; zbMATH DE number 6329100