Abstract. For all positive integers m, d, b let ρ(m,d) (resp. ρ (m,d,b)) be the maximal symmetric tensor rank of any f ∈ C[x_0, ... ,x_n] \ {0} homogeneous of degree d (resp. and with border rank ≤ b). Here we prove that ρ (m,d) ≤ \binom{m+d}{m}-m for all m ≥ 2 and d ≥ 2 (only by 1 better than a far more general result of Landsberg and Teitler), that ρ (m,d,b) ≤ ρ (b-1,d,b) if 2 ≤ b ≤ m and d ≥ b-1 and that ρ (m,d,b) ≤ d . ⌈ \binom{m+d}{m}/(m+1) ⌉ if 2 ≤ b ≤ m and 3 ≤ d ≤ b-2.

Received: March 5, 2013

AMS Subject Classification: 14N05, 14Q05, 15A69

Key Words and Phrases: symmetric tensor rank, Veronese variety, secant variety, border rank

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