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Let $f$ be a function on a finite field $F$. The decomposition of thegeneralized Fermat variety $X$ defined by the multivariate polynomial of degree$n$, $\phi(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$,plays a crucial role in the study of almost perfect non-linear (APN) functionsand exceptional APN functions. Their structure depends fundamentally on theFermat varieties corresponding to the monomial functions of exceptional degrees$n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively).Very important results for these have been obtained by Janwa, McGuire andWilson in [12,13]. In this paper we study $X$ related to the Kasami-Welchdegree monomials and its decomposition into absolutely irreducible components.We show that, in this decomposition, the components intersect transversally ata singular point.  This structural fact implies that the corresponding generalized Fermathypersurfaces, related to Kasami-Welch degree polynomial families, areabsolutely irreducible. In particular, we prove that if$f(x)=x^{2^{2k}-2^k+1}+h(x)$, where ${\rm deg}(h)\equiv 3{\pmod 4}$, then thecorresponding APN multivariate hypersurface is absolutely irreducible, andhence $f(x)$ is not exceptional APN function. We also prove conditional resultin the case when ${\rm deg}(h)\equiv 5{\pmod 8}$. Since for odd degree $f(x)$,the conjecture needs to be resolved only for the Gold degree and theKasami-Welch degree cases our results contribute substantially to the proof ofthe conjecture on exceptional APN functions---in the hardest case: theKasami-Welch degree.

On the absolute irreducibility of hyperplane sections of generalized  Fermat varieties in $\Bbb{P}^3$ and the conjecture on exceptional APN  functions: the Kasami-Welch degree case