On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in P3 and the conjecture on exceptional APN functions: the Kasami-Welch degree case

Let f be a function on a finite field F . The decomposition of the generalized Fermat variety X defined by the multivariate polynomial of degree n, φ(x, y, z) = f(x) + f(y) + f(z) in P(F2), plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees n = 2+1 and n = 22k−2k+1 (Gold and KasamiWelch numbers, respectively). Very important results for these have been obtained by Janwa, McGuire and Wilson in [12, 13]. In this paper we study X related to the Kasami-Welch degree monomials and its decomposition into absolutely irreducible components. We show that, in this decomposition, the components intersect transversally at a singular point. This structural fact implies that the corresponding generalized Fermat hypersurfaces, related to Kasami-Welch degree polynomial families, are absolutely irreducible. In particular, we prove that if f(x) = x2 2k −2+1 + h(x), where deg(h) ≡ 3 (mod 4), then the corresponding Department of Mathematics, University of Puerto Rico, Cayey Campus, San Juan PR, USA. [email protected]. Department of Mathematics, University of Puerto Rico, Rio Piedras Campus, San Juan PR, USA. [email protected]