On some Non-Permutations of Quadratic Extension of Finite Field

Bidushi Sharma; Dhiren Kumar Basnet

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arXiv · 2606.14529

On some Non-Permutations of Quadratic Extension of Finite Field

Bidushi Sharma·Dhiren Kumar Basnet

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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 $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$ 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.

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Bidushi Sharma, Dhiren Kumar Basnet. 2026-06-12. On some Non-Permutations of Quadratic Extension of Finite Field. https://arxiv.org/abs/2606.14529

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