Curves over Finite Fields and Permutations of the Form $x^k-\gamma \mathrm{Tr}(x)$

Authors : Nurdagül Anbar MEIDL

Pages : 533-538

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :We consider the polynomials of the form $Pinsert ignore into journalissuearticles values(x);=x^k-\gamma \mathrm{Tr}insert ignore into journalissuearticles values(x);$ over $\mathbb{F}_{q^n}$ for $n\geq 2$. We show that $Pinsert ignore into journalissuearticles values(x);$ is not a permutation of $\mathbb{F}_{q^n}$ in the case $\gcdinsert ignore into journalissuearticles values(k, q^n-1);>1$. Our proof uses an absolutely irreducible curve over $\mathbb{F}_{q^n}$ and the number of rational points on it.
Keywords : Function fields, permutation polynomials, rational places