We generalize to sets with cardinality more than $p$ a theorem of R\'edei and Sz\H{o}nyi on the number of directions determined by a subset $U$ of the finite plane $\mathbb F_p^2$. A $U$-rich line is a line that meets $U$ in at least $\#U/p+1$ points, while a $U$-poor line is one that meets $U$ in at most $\#U/p-1$ points. The slopes of the $U$-rich and $U$-poor lines are called $U$-special directions. We show that either $U$ is contained in the union of $n=\lceil\#U/p\rceil$ lines, or it determines `many' $U$-special directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the R\'edei polynomial to take into account the `multiplicity' of the directions determined by $U$.

中文翻译：

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由有限平面的子集决定的富贵方向

我们将 R\'edei 和 Sz\H{o}nyi 关于由有限平面 $\mathbb F_p^2$ 的子集 $U$ 确定的方向数的定理推广到基数大于 $p$ 的集合。富含 $U$ 的线是在至少 $\#U/p+1$ 点处与 $U$ 相交的线，而缺乏 $U$ 的线是在至多 $\# 处与 $U$ 相交的线U/p-1$ 点数。$U$-rich 和 $U$-poor 线的斜率称为 $U$-特殊方向。我们表明，要么 $U$ 包含在 $n=\lceil\#U/p\rceil$ 行的联合中，要么它决定了“许多”$U$-特殊方向。我们证明的核心是多项式方法的一个版本，其中我们研究了 R\'edei 多项式的迭代偏导数，以考虑由 $U$ 确定的方向的“多重性”。