Linearized polynomial maps over finite fields
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.
scientific article; zbMATH DE number 6329100
scientific article