  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 $q=2^r\geq
16$.
