Some results on permutation polynomials over finite fields
Let \(\mathbb{F}_ q\) denote the finite field of prime power order \(q=p^ e\), and let \({\mathbf X}=(x_ 1,x_ 2,\dots,x_ n)\) denote a vector of \(n\) variables. A polynomial \(g({\mathbf X})\in\mathbb{F}_ q[{\mathbf X}]\) is called a permutation polynomial (abbreviated p.p.) in \(n\) variables if and only if the equation \(g({\mathbf X})=a\) has \(q^{n-1}\) solutions for each \(a\in\mathbb{F}_ q\). Now let \({\mathbf X}\) be partitioned as \({\mathbf X}=({\mathbf X}_ 1,{\mathbf X}_ 2,\dots,{\mathbf X}_ t)\) where for each \(i=1,2,\dots,t\), \({\mathbf X}_ i\) is a vector of \(v_ i\) variables, \(v_ 1+v_ 2+\cdots+v_ t=n\). Further, let \(f_ i({\mathbf X}_ i)\) denote a polynomial in \(v_ i\) variables over \(\mathbb{F}_ q\), and let \(f({\mathbf X})\) be the polynomial defined by \(f({\mathbf X})=f_ 1({\mathbf X}_ 1)+f_ 2({\mathbf X}_ 2)+\cdots+f_ t({\mathbf X}_ t)\). This paper derives necessary and sufficient conditions in order that \(f({\mathbf X})\) be a p.p. in \(n\) variables. More specifically, the author shows that \(f({\mathbf X})\) is a p.p. if and only if for every order \(p^{e-1}\) subgroup \(H\) of the additive group \(G=(\mathbb{F}_ q,+)\), some one of the \(f_ i({\mathbf X}_ i)\) distributes the members of \(F_ q^{v_ i}\) uniformly over the cosets of \(H\) in \(G\). This result specializes to previously known results obtained by \textit{R. Lidl} and \textit{H. Niederreiter} [Acta Arith. 22, 257-265 (1973; Zbl 0261.12012)]. The present paper also gives necessary and sufficient conditions in order that a polynomial in \(n\) variables be a p.p. on the Galois ring \(GR(p^ e,m)\).
scientific article; zbMATH DE number 91063