Linear restrictions on cone polynomials

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For a set S of d points in the n-dimensional projective space over a field of characteristic zero, we prove that the polynomials of degree d whose zero sets are cones over S do not span the vector space of polynomials of degree d vanishing on S, if d is odd and d≥ 3. Furthermore, they span a subspace of codimension at least two, if n=2, d=1 4 and d≥ 5.