Nonnegative polynomials and their Carathéodory number

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In 1888 Hilbert showed that every nonnegative homogeneous polynomial with real coefficients of degree 2d in n variables is a sum of squares if and only if d=1 (quadratic forms), n=2 (binary forms) or (n,d)=(3,2) (ternary quartics). In these cases, it is interesting to compute canonical expressions for these decompositions. Starting from Carathéodory's Theorem, we compute the Carathéodory number of Hilbert cones of nonnegative quadratic and binary forms.