Unique decomposition for a polynomial of low rank

Edoardo Ballico, and Alessandra Bernardi. "Unique decomposition for a polynomial of low rank." Annales Polonici Mathematici 108.3 (2013): 219-224. <http://eudml.org/doc/280239>.

@article{EdoardoBallico2013,
abstract = {Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of $ℙ^m$ into $ℙ^\{\{m+d \atop d\}-1\}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F = M₁^d + ⋯ + M_t^d + Q$, where $M₁,. .., M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = ∑_\{i=1\}^q l_i^\{d-d_i\} m_i$ with $l_i$’s linear forms and $m_i$’s forms of degree $d_i$ such that $∑(d_i + 1) = s - t.$},
author = {Edoardo Ballico, Alessandra Bernardi},
journal = {Annales Polonici Mathematici},
keywords = {Waring problem; polynomial decomposition; symmetric rank; symmetric tensors; Veronese varieties; secant varieties},
language = {eng},
number = {3},
pages = {219-224},
title = {Unique decomposition for a polynomial of low rank},
url = {http://eudml.org/doc/280239},
volume = {108},
year = {2013},
}

TY - JOUR
AU - Edoardo Ballico
AU - Alessandra Bernardi
TI - Unique decomposition for a polynomial of low rank
JO - Annales Polonici Mathematici
PY - 2013
VL - 108
IS - 3
SP - 219
EP - 224
AB - Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of $ℙ^m$ into $ℙ^{{m+d \atop d}-1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F = M₁^d + ⋯ + M_t^d + Q$, where $M₁,. .., M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = ∑_{i=1}^q l_i^{d-d_i} m_i$ with $l_i$’s linear forms and $m_i$’s forms of degree $d_i$ such that $∑(d_i + 1) = s - t.$
LA - eng
KW - Waring problem; polynomial decomposition; symmetric rank; symmetric tensors; Veronese varieties; secant varieties
UR - http://eudml.org/doc/280239
ER -