An upper bound for the Waring rank of a form

TL;DR

This paper introduces the open Waring rank of a form and establishes a new upper bound for the classical Waring rank, improving previous results and providing the best known upper bound.

Contribution

It defines the open Waring rank and proves a new, tighter upper bound for the classical Waring rank of forms in multiple variables.

Findings

Established an upper bound for the open Waring rank.

Proved the same bound applies to the classical Waring rank.

Improved upon previous upper bound results.

Abstract

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 \binom{n+d-2}{d-1} - \binom{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 [BBS] and giving, as far as we know, the best upper bound known.