On two conjectures about the intersection distribution

Abstract

Recently, Li and Pott proposed a new concept of intersection distribution concerning the interaction between the graph \(\{(x,f(x))~|~x\in {\mathbb {F}}_{q}\}\) of f and the lines in the classical affine plane AG(2, q). Later, Kyureghyan et al. proceeded to consider the next simplest case, and derived the intersection distribution for all degree three polynomials over \({\mathbb {F}}_{q}\) with q both odd and even. They also proposed several conjectures therein. In this paper, we completely solve two conjectures of Kyureghyan et al. Namely, we prove two classes of power functions having intersection distribution: \(v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}\). We mainly make use of the multivariate method and a certain type of equivalence on 2-to-1 mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations.