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Register a new userOn the existence of dimension zero divisors in algebraic function fields defined over F_q
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Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree gamma-1 where gamma is the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.
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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 $q=2^rgeq 16$.
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