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Mathematische Zeitschrift

, Volume 271, Issue 3–4, pp 1141–1149 | Cite as

Decomposition of homogeneous polynomials with low rank

  • Edoardo BallicoEmail author
  • Alessandra Bernardi
Article

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 s-th secant variety of the d-uple Veronese embedding of \({\mathbb{P}^m}\) into \({\mathbb{P}^{\tiny\left(\begin{array}{c}{\rm m+d} \\ {\rm d}\end{array}\right)-1}}\) but that its minimal decomposition as a sum of d-th powers of linear forms M 1, . . . , M r is \({F=M_1^d+\cdots + M_r^d}\) with r > s. We show that if s + r ≤ 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.

Keywords

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

Mathematics Subject Classification (2000)

15A21 15A69 14N15 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly
  2. 2.GALAAD, INRIA MéditerranéeSophia AntipolisFrance

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