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

2004/10/06 by Stephane Ballet, Ballet, Stephane, Dominique Le Brigand +1 · 3 citations
Mathematics · #11R58 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11R58

paper · pdf · doi:10.48550/arxiv.math/0410193

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

arxiv created 2004/10/13 · arxiv updated 2009/12/01

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_qⁿ of \F_q, when q=2^r≥ 16.

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