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

Abstract We study the existence of non-special divisors of degree g and g - 1 for algebraic function fields of genus g ⩾ 1 defined over a finite field F q . In particular, we prove that there always exists an effective non-special divisor of degree g ⩾ 2 if q ⩾ 3 and that there always exists a non-special divisor of degree g - 1 ⩾ 1 if q ⩾ 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 ⩾ 16 .