Curvilinear schemes and maximum rank of forms

Abstract

We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$-th Veronese embedding of ${\mathbb{P}}^n$, whose span contains the projective class of $P$. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.

Changed Questions 2 and 3. More detailed proofs

References in corpus (4)

• Computing symmetric rank for symmetric tensors
• Ranks of tensors and a generalization of secant varieties
• On the cactus rank of cubics forms
• Stratification of the fourth secant variety of Veronese variety via the symmetric rank

Cited by in corpus (1)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition