Ballico, Edoardo ; Bernardi, Alessandra
(2010)
*Decomposition of homogeneous polynomials with low rank.*
[Preprint]

*This is the most updated version of the document.*

Full text available as:

## 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 $M_1, \ldots , M_r$ is $F=M_1^d+\cdots + M_r^d$ 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.

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 $M_1, \ldots , M_r$ is $F=M_1^d+\cdots + M_r^d$ 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.

Document type

Preprint

Creators

Keywords

Waring problem, Polynomial decomposition, Symmetric rank, Symmetric tensors, Veronese varieties, Secant varieties.

Subjects

DOI

Deposit date

01 Sep 2010 08:21

Last modified

16 May 2011 12:14

URI

## Other metadata

Document type

Preprint

Creators

Keywords

Waring problem, Polynomial decomposition, Symmetric rank, Symmetric tensors, Veronese varieties, Secant varieties.

Subjects

DOI

Deposit date

01 Sep 2010 08:21

Last modified

16 May 2011 12:14

URI

### Available versions of this document

This work may be freely consulted and used, may be reproduced on a permanent basis in a digital format (i.e. saving) and can be printed on paper with own personal equipment (without availing of third -parties services), for strictly and exclusively personal, research or teaching purposes, with express exclusion of any direct or indirect commercial use, unless otherwise expressly agreed between the user and the author or the right holder. It is also allowed, for the same purposes mentioned above, the retransmission via telecommunication network, the distribution or sending in any form of the work, including the personal redirection (e-mail), provided it is always clearly indicated the complete link to the page of the Alma DL Site in which the work is displayed. All other rights are reserved.
## Downloads

## Downloads

Staff only: