标题： 关于两个关于交分布的猜想 显示英文标题

标题： On two conjectures about the intersection distribution

摘要： 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. 他们还在\cite{KLP}中提出了几个猜想。 在本文中，我们完全解决了\cite{KLP}中的两个猜想。 即，我们证明了两类具有交集分布的幂函数：$v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}$。 我们主要利用了多变量方法和$2$到$1$映射上的 QM 等价性。 我们证明的关键在于考虑某些低次方程的解的数量。 显示英文摘要

摘要： 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.