On $\mathbb{F}_q$-primitive points on hypersurfaces

TL;DR

This work investigates the count of F_q-primitive points on affine hypersurfaces defined by f(x_1,...,x_s)=0, introducing P_q(f) as the number of F_q-rational points with all coordinates primitive. It develops a general framework expressing P_q(f) through multiplicative character sums and inclusion-exclusion, proving sharp bounds for Dwork-regular polynomials and for Fermat-type polynomials via Katz-type mixed sums and Jacobi sums, respectively. A Cohen-Huczynska sieve is employed to obtain existence results and to prove a conjecture from Takshak et al., while the Fermat-prime case yields an explicit formula for P_q on affine hyperplanes, including an exact result for linear forms. The results advance understanding of primitive-coordinate points on higher-dimensional varieties and connect primitive-element problems to Deligne/Dwork-regularity, Jacobi sums, and sieve methods, with explicit consequences in the Fermat-prime setting.

Abstract

In this paper, we estimate the number of $\mathbb{F}_q$-primitive points on the affine hypersurface defined by the equation $f(x_1,\ldots,x_s)=0$, where $f\in\mathbb{F}_q[x_1,\dots,x_s]$ is an appropriate polynomial. In particular, we provide existence results for the case when $f$ is Dwork-regular and when $f$ is of Fermat type. Additionally, we present a proof for a recently posed conjecture. Finally, in the case where $q$ is a Fermat prime, we provide an explicit formula for the number of $\mathbb{F}_q$-primitive points on hyperplanes.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Example 7
• Lemma 8
• Lemma 9
• Lemma 10: Lema 3.5, ReisRibas