Decomposition of homogeneous polynomials with low rank - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Mathematische Zeitschrift Année : 2012

Decomposition of homogeneous polynomials with low rank

Résumé

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 $s$-th secant variety of the $d$-uple Veronese embedding of $\mathbb{P}^m$ into $ \PP {{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 $r$ is at most $d$ and a mild condition is satisfied.
Fichier principal
Vignette du fichier
pollo2.pdf (252.25 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00645978 , version 1 (28-11-2011)

Identifiants

Citer

Edoardo Ballico, Alessandra Bernardi. Decomposition of homogeneous polynomials with low rank. Mathematische Zeitschrift, 2012, 271, pp.1141-1149. ⟨10.1007/s00209-011-0907-6⟩. ⟨hal-00645978⟩
160 Consultations
181 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More