On two conjectures about the intersection distribution

10/01/2020

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by   Yubo Li, et al.

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Recently, S. Li and A. Pott<cit.> proposed a new concept of intersection distribution concerning the interaction between the graph {(x,f(x)) | x∈_q} of f and the lines in the classical affine plane AG(2,q). Later, G. Kyureghyan, et al.<cit.> proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over _q with q both odd and even. They also proposed several conjectures in <cit.>. In this paper, we completely solve two conjectures in <cit.>. Namely, we prove two classes of power functions having intersection distribution: v_0(f)=q(q-1)/3, v_1(f)=q(q+1)/2, v_2(f)=0, v_3(f)=q(q-1)/6. We mainly make use of the multivariate method and QM-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.