On some Non-Permutations of Quadratic Extension of Finite Field

Abstract

In this article, we study polynomials over ${\mathbb{F}}_{q^2}$ that do not permute ${\mathbb{F}}_{q^2}$. More precisely, we characterize polynomials of the forms $x^q+b{x}^2+cx+d$ and $x^{q+1}+b{x}^q+cx+d$ over ${\mathbb{F}}_{q^2}$ according to whether they are permutation or non-permutation polynomials. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums.

14 pages