Counting points of homogeneous varieties over finite fields

TL;DR

This paper proves that for algebraic varieties homogeneous under a linear algebraic group over finite fields, the count of rational points over extensions is a periodic polynomial with non-negative coefficients after a shift.

Contribution

It establishes the periodic polynomial nature of point counts for homogeneous varieties over finite fields and shows the shifted version has non-negative coefficients.

Findings

Number of rational points is a periodic polynomial in $q^n$.

Shifted polynomial with $q^n + 1$ has non-negative coefficients.

Results apply to varieties homogeneous under linear algebraic groups.

Abstract

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer coefficients. Moreover, the shifted periodic polynomial function, where $q^n$ is formally replaced with $q^n + 1$, is shown to have non-negative coefficients.