  Let C be a real nonsingular affine curve of genus one, embedded in affine
n-space, whose set of real points is compact. For any polynomial f which is
nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv
\sum_i f_i^2 (modulo I_C) and such that the degrees deg(f_i) are bounded in
terms of deg(f) only. Using Lasserre's relaxation method, we deduce an explicit
representation of the convex hull of C(R) in R^n 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.
