On the existence of dimension zero divisors in algebraic function fields defined over F_q

TL;DR

This paper investigates the existence and properties of dimension zero divisors in algebraic function fields over finite fields, providing new existence results, conditions, and density estimates.

Contribution

It offers new results on the existence, number, and density of dimension zero divisors, including specific conditions for hyperelliptic fields and cases over small finite fields.

Findings

Existence of dimension zero divisors of degree g-1 for q=2,3

Necessary and sufficient conditions for hyperelliptic fields

Density estimates of such divisors

Abstract

Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree \gamma-1 where \gamma is the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.