Curvilinear schemes and maximum rank of forms

TL;DR

This paper introduces the concept of curvilinear rank for homogeneous polynomials and establishes bounds on their rank based on this new measure, linking geometric schemes to polynomial complexity.

Contribution

It defines curvilinear rank for forms and provides bounds on polynomial rank depending on this new geometric invariant.

Findings

Defined curvilinear rank for degree d forms.

Established bounds relating polynomial rank to curvilinear rank.

Connected geometric schemes with polynomial decomposition complexity.

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.