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Let X be an algebraic curve, defined over a perfect field, and G a divisor onX. If X has sufficiently many points, we show how to construct a divisor D on Xsuch that l(2D-G)=0, of essentially any degree such that this is compatible theRiemann-Roch theorem. We also generalize this construction to the case of afinite number of constraints, l(k_i.D-G_i)=0, where |k_i|\leq 2.  Such a result was previously claimed by Shparlinski-Tsfasman-Vladut, inrelation with the Chudnovsky-Chudnovsky method for estimating the bilinearcomplexity of the multiplication in finite fields based on interpolation oncurves; unfortunately, as noted by Cascudo et al., their proof was flawed. Soour work fixes the proof of Shparlinski-Tsfasman-Vladut and shows that theirestimate m_q\leq 2(1+1/(A(q)-1)) holds, at least when A(q)\geq 5. We also fix astatement of Ballet that suffers from the same problem, and then we point out afew other possible applications.

Diviseurs de la forme 2D-G sans sections et rang de la multiplication  dans les corps finis (Divisors of the form 2D-G without sections and bilinear  complexity of multiplication in finite fields)