Further Perspectives on the Intersection Distribution

Abstract: The concepts of intersection distribution and non-hitting index were recently introduced by Li and Pott, and offer a new way to classify the behaviour of finite field polynomials. They have both an algebraic and geometric interpretation: via the intersection of a polynomial $f$ with a set of lines, and via a $(q+1)$-set $S_f$ in $\mathrm{PG}(2,q)$ possessing an internal nucleus. In this paper, we build on these ideas: we prove novel geometric results (particularly on the relationship between intersection distribution and projective equivalence of polynomials), new algebraic results (particularly on the degree of $S_f$ - the index of the largest non-zero entry in the intersection distribution of $f$) and new results on the non-hitting spectrum. We resolve several Open Problems from Li and Pott's original paper, and offer alternative treatments of related subsequent literature.