# Intersections between the norm-trace curve and some low degree curves

## TL;DR

This paper studies the intersection points between the norm-trace curve over a finite field and certain low-degree curves, providing detailed characterizations and bounds that aid in understanding related algebraic geometry codes.

## Contribution

It offers a complete characterization of intersections with parabolas and bounds for other low-degree curves, advancing the analysis of algebraic geometry codes.

## Key findings

- Complete intersection characterization with parabolas
- Sharp bounds for intersections with other low-degree curves
- Application to weight distribution of AG codes

## Abstract

In this paper we analyze the intersection between the norm-trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, giving a complete characterization of the intersection between the curve and the parabolas, as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes constructed on the curve.