On the existence of non-special divisors of degree $g$ and $g-1$ in algebraic function fields over $\F_q$

TL;DR

This paper proves the existence of non-special divisors of certain degrees in algebraic function fields over finite fields and applies these results to improve bounds on the bilinear complexity of multiplication in field extensions.

Contribution

It establishes new existence results for non-special divisors of degrees g and g-1 in algebraic function fields over finite fields, leading to improved complexity bounds.

Findings

Existence of effective non-special divisors of degree g for q ≥ 3

Existence of non-special divisors of degree g-1 for q ≥ 4

Improved upper bounds on bilinear complexity of multiplication in extensions of finite fields

Abstract

We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $g\geq 2$ if $q\geq 3$ and that there always exists a non-special divisor of degree $g-1\geq 1$ if $q\geq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $\F_{q^n}$ of $\F_q$, when $q=2^r\geq 16$.