An upper bound for the Waring rank of a form

AbstractIn this paper we introduce the open Waring rank of a form of degree d in n variables and prove that this rank is bounded from above by $$\binom{n + d - 2}{d - 1} - \binom{n + d - 6}{d - 3}$$n+d-2d-1-n+d-6d-3whenever n, d ≥ 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of Białynicki-Birula and Schinzel (see[4]) and giving, as far as we know, the best upper bound known.