Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

TL;DR

This paper investigates Frobenius nonclassical curves over finite fields with respect to linear systems of arbitrary degree, providing explicit conditions, point counts, and bounds based on the degree and Frobenius properties.

Contribution

It introduces a family of Frobenius nonclassical curves for any degree s and characterizes their properties, including rational point counts and conditions for nonclassicality.

Findings

Explicit family of Frobenius nonclassical curves for each degree s

Necessary and sufficient conditions for s=2 case

Exact rational point counts in nonclassical cases

Abstract

For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for such curves to be $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of conics. In the $\mathbb{F}_q$-Frobenius nonclassical cases, we determine the exact number of $\mathbb{F}_q$-rational points. In the remaining cases, an upper bound for the number of $\mathbb{F}_q$-rational points will follow from St\"ohr-Voloch theory.