Abstract
Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | ≤ ( n + ⌊ d / 2 ⌋ n ) . For the tensor rank, we describe the cases with | S | ≤ 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d).
Highlights
Let X ⊂ Pr be an integral and non-degenerate variety
For any q ∈ Pr, the X-rank r X (q) of q is the minimal cardinality of a finite set S ⊂ X such that q ∈ hSi, where h i denotes the linear span
The definition of X-ranks captures the notion of tensor rank of rank decomposition of a homogeneous polynomial of partially symmetric tensor rank and small variations of it may be adapted to cover other applications
Summary
Let X ⊂ Pr be an integral and non-degenerate variety. For any q ∈ Pr , the X-rank r X (q) of q is the minimal cardinality of a finite set S ⊂ X such that q ∈ hSi, where h i denotes the linear span. Let S ⊂ X be a finite set and q ∈ Pr. We say that S evinces the X-rank of q if q ∈ hSi and. S may evince an X-rank only if it is linearly independent, but this condition is not a sufficient one, except in very trivial cases, like when r X (q) ≤ 2 for all q ∈ Pr. Call r X,max the maximum of all integers r X (q). If |S| ≤ b(ρ( X ) + 1)/2c, S totally evinces the X-ranks (as in [43] Theorem 1.18) while, for each integer t > b(ρ( X ) + 1)/2c with t ≤ r + 1, there is a linearly independent subset of X with cardinality t and not totally evincing the X-ranks ( Lemma 3).
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