The second largest number of points of plane curves over finite fields

TL;DR

This paper determines the second largest number of rational points on plane curves of degree d over finite fields, providing a basis for the ideal of the complement of a linear subspace in projective space.

Contribution

It introduces a basis for the ideal of the complement of a linear subspace in projective space over finite fields and applies this to find the second largest number of points on plane curves.

Findings

Identifies the second largest number of points on plane curves of degree d over finite fields.

Provides a basis for the ideal of the complement of a linear subspace in projective space.

Applies algebraic geometry techniques to finite field point counting.

Abstract

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is also given for $d\geq q+1$.