# On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

## TL;DR

This paper surveys the tensor rank of multiplication in finite field extensions, discusses recent and unpublished results, and explores connections to open problems in number theory, algebraic geometry, and coding theory.

## Contribution

It provides a comprehensive overview of known results, clarifies unresolved issues, and links tensor rank problems to broader mathematical areas.

## Key findings

- Summarizes known results on tensor rank in finite fields
- Highlights unresolved problems and recent unpublished results
- Connects tensor rank issues to open problems in algebraic geometry and number theory

## Abstract

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative understanding of the domain. In particular, we identify and clarify certain results not completely proved and we emphasis the link with open problems in number theory, algebraic geometry, and coding theory.