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LetX ⊂  P rbe an integral and non-degenerate variety. We study when a finite setS ⊂ Xevinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embeddingX  n , dofP nand| S |  ≤ (   n + ⌊ d / 2 ⌋  n  )    . For the tensor rank, we describe the cases with| S | ≤ 3    . ForX  n , d   ,we raise some questions of the maximum rank ford ≫ 0(for a fixed n) and forn ≫ 0(for a fixed d).

Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank