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arXiv · 1811.12631

The Weil bound and non-exceptional permutation polynomials over finite fields

Xiang Fan
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Abstract

A well-known result of von zur Gathen asserts that a non-exceptional permutation polynomial of degree $n$ over $\mathbb{F}_{q}$ exists only if $q<n^{4}$. With the help of the Weil bound for the number of $\mathbb{F}_{q}$-points on an absolutely irreducible (possibly singular) affine plane curve, Chahal and Ghorpade improved von zur Gathen's proof to replace $n^{4}$ by a bound less than $n^{2}(n-2)^{2}$. Also based on the Weil bound, we further refine the upper bound for $q$ with respect to $n$, by a more concise and direct proof following Wan's arguments.

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Xiang Fan. 2018-11-30. The Weil bound and non-exceptional permutation polynomials over finite fields. https://arxiv.org/abs/1811.12631

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