For a positive integer k and a linearized polynomial L(X), polynomials of the form \(P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]\) are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of \(\mathbb {F}_{q^{n}}\), then P(X) cannot be a permutation if \(\gcd (k,q^{n}-1)>1\). Further, necessary conditions for P(X) to be a permutation of \(\mathbb {F}_{q^{n}}\) are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of \(\mathbb {F}_{q^{n}}\), and their number of rational affine points.

对于正整数k和线性多项式L（X），在{\ mathbb F} _ {q ^ {中，\（P（X）= G（X）^ {k} -L（X）\\ n}} [X] \）。结果表明，当大号具有一个非平凡内核和G ^是的置换\（\ mathbb {F} _ {Q ^ {N}} \） ，则P（X）不能置换如果\（\ GCD （k，q ^ {n} -1）> 1 \）。此外，对于G（X的情况，给出了P（X）为\（\ mathbb {F} _ {q ^ {n}} \）的置换的必要条件。）是任意线性化多项式。该方法使用的平面曲线是通过\（\ mathbb {F} _ {q ^ {n}} \）的乘法和加法结构及其有理仿射点数获得的。