Decomposition of homogeneous polynomials with low rank

TL;DR

This paper studies the decomposition of homogeneous polynomials with low rank, showing conditions under which such decompositions can be split into simpler parts and establishing uniqueness results for minimal decompositions.

Contribution

It introduces a method to split polynomial decompositions into parts involving only two variables and proves uniqueness of minimal decompositions under certain rank conditions.

Findings

Decomposition can be split into a two-variable part and a uniquely determined remaining part.

Uniqueness of minimal decomposition is established for ranks up to the degree d.

Conditions for splitting and uniqueness depend on the sum of the secant variety rank and the decomposition length.

Abstract

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1, ..., M_r$ is $F=M_1^d+... + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.