论文标题
关于交叉分布的两个猜想
On two conjectures about the intersection distribution
论文作者
论文摘要
最近，S。Li和A. Pott \ cite {lp}提出了一个有关图形$ \ {（x，x，f（x））〜| 〜x \ in \ f_ {q} \ iN $ f $的相互作用的新概念，$ \ {（x，f（x））后来，G。Kyureghyan等人。他们还提出了\ cite {klp}的几个猜想。 在本文中，我们在\ cite {klp}中完全解决了两个猜想。也就是说，我们证明了具有相交分布的两类幂函数： $ v_ {0}（f）= \ frac {q（q-1）} {3}，〜v_ {1}（f）= \ frac {q（q+1）} {2} {2}，〜V_ {2}（2}（f）（f）= 0，〜V_ v_ {3}（3}（f）= \ frac = \ frac = \ frac {q（q（q）我们主要利用多元方法和QM等效性，价格为$ 2 $ -to- $ 1 $映射。我们证明的关键是考虑一些低度方程的解决方案的数量。
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.
