On the non-special divisors in algebraic function fields defined over $\mathbb{F}_q$

TL;DR

This paper surveys known results and introduces new findings on the existence of non-special divisors in algebraic function fields over finite fields, emphasizing their importance in applications like information theory.

Contribution

It provides a comprehensive, self-contained survey with full proofs and new results on non-special divisors in algebraic function fields over finite fields.

Findings

Existence results for non-special divisors in curves of defect k

Full proofs of known theorems and new existence criteria

Enhanced understanding of divisors' role in algebraic function fields

Abstract

In the theory of algebraic function fields and their applications to the information theory, the Riemann-Roch theorem plays a fundamental role. But its use, delicate in general, is efficient and practical for applications especially in the case of non-special divisors. So, in this paper, we give a survey of the known results concerning the non-special divisors in the algebraic function fields defined over finite fields, enriched with some new results about the existence of such divisors in curves of defect k. In particular, we have chosen to be self-contained by giving the full proofs of each result, the original proofs or shorter alternative proofs.