An upper bound for the Waring rank of a form

摘要

In 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(n + d ? 2 d ? 1)? (n + d ? 6 d ? 3) whenever n, d ≥ 3. This proves the same upper bound for the classical Waring rank of a form, improving the result of Bia?lynicki-Birula and Schinzel (see [4]) and giving, as far as we know, the best upper bound known.