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)

Let X be an algebraic curve, defined over a perfect field, and G a divisor on X. If X has sufficiently many points, we show how to construct a divisor D on X such that l(2D-G)=0, of essentially any degree such that this is compatible the Riemann-Roch theorem. We also generalize this construction to the case of a finite 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, in relation with the Chudnovsky-Chudnovsky method for estimating the bilinear complexity of the multiplication in finite fields based on interpolation on curves; unfortunately, as noted by Cascudo et al., their proof was flawed. So our work fixes the proof of Shparlinski-Tsfasman-Vladut and shows that their estimate m_q\leq 2(1+1/(A(q)-1)) holds, at least when A(q)\geq 5. We also fix a statement of Ballet that suffers from the same problem, and then we point out a few other possible applications.

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