Geometric characterization of p-exceptional monomial GAPN functions

On finite fields of characteristic p, PN (perfect nonlinear) functions for odd

p and APN (almost perfect nonlinear) functions for even p are well-known

classes of highly nonlinear functions. GAPN (generalized almost perfect

nonlinear) functions were introduced as a generalization of APN functions for

even p to all p. One of the main targets of studies on such highly nonlinear

functions is their classification. While p-exceptional monomial PN and 2-

exceptional monomial APN functions have been classified, the corresponding

problem for GAPN functions remains open. Here, a polynomial over Fp is

called a p-exceptional PN (resp. APN, resp. GAPN) function if it is a PN

(resp. APN, resp. GAPN) function on Fpn for infinitely many positive integers n.

In this talk, we give a geometric characterization of p-exceptional mono-

mial GAPN functions. To this end, for a prime power q, we establish a

geometric characterization of non-zero polynomials in Fq[x, y] having no ab-

solutely irreducible factor defined over Fq.

Related preprint: https://arxiv.org/pdf/2608.08969