Partially complex ranks for real projective varieties

Let X((C) subset of Pr((C) be an integral non-degenerate variety defined over. For any q epsilon P-r (C) we study the existence of S subset of X((C) with small cardinality, invariant for the complex conjugation and with q contained in the real linear space spanned by S. We discuss the advantages of these additive decompositions with respect to the X(R) rank, i.e. the rank of q with respect to X(R). We describe the case of hypersurfaces and Veronese varieties.