An upper bound for the Waring rank of a form

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In this paper we introduce the open Waring rank of a form of degree d in n variables and prove the that this rank in bounded from above by n+d-2d-1 - n+d-6d-3 whenever n, d >= 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of [BBS] and giving, as far as we know, the best upper bound known.