Convex hulls of curves of genus one - Mathematics > Algebraic GeometryReport as inadecuate

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Abstract: Let C be a real nonsingular affine curve of genus one, embedded in affinen-space, whose set of real points is compact. For any polynomial f which isnonnegative on CR, we prove that there exist polynomials f i with f \equiv\sum i f i^2 modulo I C and such that the degrees degf i are bounded interms of degf only. Using Lasserre-s relaxation method, we deduce an explicitrepresentation of the convex hull of CR in R^n by a lifted linear matrixinequality. This is the first instance in the literature where such arepresentation is given for the convex hull of a nonrational variety. The sameworks for convex hulls of singular curves whose normalization is C. We thenmake a detailed study of the associated degree bounds. These bounds aredirectly related to size and dimension of the projected matrix pencils. Inparticular, we prove that these bounds tend to infinity when the curve Cdegenerates suitably into a singular curve, and we provide explicit lowerbounds as well.

Author: Claus Scheiderer


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