Type of Publication:  Journal article 
Author:  Scheiderer, Claus 
Year of publication:  2011 
Published in:  Advances in Mathematics ; 228 (2011), 5.  pp. 26062622.  ISSN 00018708 
DOI (citable link):  https://dx.doi.org/10.1016/j.aim.2011.07.014 
Summary: 
Let C be a real nonsingular affine curve of genus one, embedded in affine nspace, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials fi with View the MathML source (mod IC) and such that the degrees deg(fi) are bounded in terms of deg(f) only. Using Lasserreʼs relaxation method, we deduce an explicit representation of the convex hull of C(R) in Rn by a lifted linear matrix inequality. This is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety. The same works for convex hulls of (singular) curves whose normalization is C. We then make a detailed study of the associated degree bounds. These bounds are directly related to size and dimension of the projected matrix pencils. In particular, we prove that these bounds tend to infinity when the curve C degenerates suitably into a singular curve, and we provide explicit lower bounds as well.

Subject (DDC):  510 Mathematics 
Keywords:  Real algebraic curves, convex hulls, elliptic curves, linear matrix inequalities, spectrahedra, Lasserre relaxation 
Bibliography of Konstanz:  Yes 
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SCHEIDERER, Claus, 2011. Convex hulls of curves of genus one. In: Advances in Mathematics. 228(5), pp. 26062622. ISSN 00018708. Available under: doi: 10.1016/j.aim.2011.07.014
@article{Scheiderer2011Conve19147, title={Convex hulls of curves of genus one}, year={2011}, doi={10.1016/j.aim.2011.07.014}, number={5}, volume={228}, issn={00018708}, journal={Advances in Mathematics}, pages={26062622}, author={Scheiderer, Claus} }
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