On two conjectures about the intersection distribution

TL;DR

This paper proves two conjectures regarding the intersection distribution of certain power functions over finite fields, using multivariate methods and solution counting of low-degree equations, advancing understanding of polynomial interactions with affine lines.

Contribution

It completely solves two conjectures about intersection distributions for specific power functions over finite fields, employing novel multivariate and QM-equivalence techniques.

Findings

Identified classes of power functions with specific intersection distributions.

Used multivariate methods and solution counting to prove conjectures.

Provided explicit intersection distribution formulas for these functions.

Abstract

Recently, S. Li and A. Pott\cite{LP} proposed a new concept of intersection distribution concerning the interaction between the graph $\{(x,f(x))~|~x\in\F_{q}\}$ of $f$ and the lines in the classical affine plane $AG(2,q)$. Later, G. Kyureghyan, et al.\cite{KLP} proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\F_{q}$ with $q$ both odd and even. They also proposed several conjectures in \cite{KLP}. In this paper, we completely solve two conjectures in \cite{KLP}. 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 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.