On the Decomposition of Generalized Fermat Varieties in $P^3$ Corresponding to Kasami-Welch Functions

Abstract

The study of the generalized Fermat variety

$${\varphi }_j=\frac{x^j+y^j+z^j+(x+y+z{)}^j}{(x+y)(x+z)(y+z)}$$

defined over a finite field $L={\mathbb{F}}_q$, where $q=2^n$ for some positive integer $n$, plays an important role in the study of (APN) functions and exceptional APN functions. This study arose after a characterization by Rodier that relates these functions with the number of rational points of ${\varphi }_j=(x,y,z)$. The most studied cases are when $j=2^k+1$ and $j=2^{2k}–2^k+1$, the Gold and Kasami-Welch numbers. In this article, we make a claim about the decomposition of ${\varphi }_j$ into absolutely irreducible components. If these components intersect transversally at a particular point, then the corresponding Kasami-Welch polynomial is absolutely irreducible. This implies that the function is not exceptional APN, thus helping us make progress on the stated conjecture.