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

We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $ggeq 1$ defined over a finite field $F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $ggeq 2$ if $qgeq 3$ and that there always exists a non-special divisor of degree $g-1geq 1$ if $qgeq 4$. 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 $q=2^rgeq 16$.