Intersection distribution of degree four polynomials over finite fields

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.