APN monomials over GF(n2) for infinitely many n

I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over Fn2 for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, in: Actualités Sci. Ind., vol. 1041, Hermann, Paris, 1948] on absolutely irreducible curves shows that a monomial xm is not APN over Fn2 for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over F2. I will show that the latter polynomial's singularities imply that except in three specific, narrowly defined cases, all monomials have such a factor over a finite field of characteristic 2. Two of these cases, those with exponents of the form k2+1 or k4−k2+1 for any integer k, are already known to be APN for infinitely many fields. The last, relatively rare case when a certain gcd is maximal is still unproven; my method fails. Some specific, special cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.