Given a polynomial f over the finite field \(\mathbb {F}_q\), its intersection distribution provides fruitful information on how lines in the affine plane intersect the graph of f over \(\mathbb {F}_q\). The intersection distribution in its simplest cases gives rise to oval polynomials in finite geometry and Steiner triple systems in design theory. Previously, the intersection distribution of degree two and degree three polynomials has been computed. In this paper, we determine the intersection distribution of all degree four polynomials over finite fields. As an application, we present a direct construction of Steiner systems using polynomials with prescribed intersection distribution.

给定有限域 \（\mathbb {F}_q\） 上的多项式 f，它的交集分布提供了关于仿射平面中的线如何在 \（\mathbb {F}_q\） 上与 f 图相交的丰富信息。在最简单的情况下，交集分布在有限几何中产生了椭圆多项式，在设计理论中产生了 Steiner 三元组。以前，已经计算了 2 次和 3 次多项式的交集分布。在本文中，我们确定了有限域上所有四次多项式的交集分布。作为一个应用，我们提出了一个使用具有指定交集分布的多项式的 Steiner 系统的直接构造。