Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support

Let , , denote the degree Veronese embedding of . For any , let be the minimal cardinality of such that . Identifying with a homogeneous polynomial (or a symmetric tensor), corresponds to writing as a sum of powers with a linear form (or as a sum of -powers of vectors). Here we fix an integral variety and and study a similar decomposition with and minimal. For instance, if is a linear subspace, then we prove that and classify all such that .