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

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

2004-10-06

arxiv.org/abs/math/0410193v2

Mathematics

Number Theory

21 pages: added Remark 22 at the end of the paper

Scientific paper

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 $q=2^r\geq 16$.

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