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

Abstract

We consider the polynomials of the form $P(X)=x^k-gamma Tr(x)$ over $F_qn$ for $ngeq2$ We show that P(x) is not a permutation of $F_qn$ in the case $gcd(k,;q^n-1)>1$.Our proof uses an absolutely irreducible curve over $F_qn$ andthe number of rational points on it.