Vandermonde sets, hyperovals and Niho bent functions

Abstract

<p style='text-indent:20px;'>We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of <inline-formula><tex-math id="M1">\begin{document}$ g $\end{document}</tex-math></inline-formula>-functions, which can be considered as an analog of Glynn's Theorem for <inline-formula><tex-math id="M2">\begin{document}$ o $\end{document}</tex-math></inline-formula>-polynomials. We also get some important applications to Niho bent functions.</p>