Decomposition of homogeneous polynomials with low rank

By Edoardo Ballico, Alessandra Bernardi. 2010

85 citations recorded by the source corpus.

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 M1, . . . , Mr 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.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0a483021c6aa54260b3fa94bc51738625c2d266e

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In Computational Mathematics

  1. Alessandra Bernardi
  2. A. Oneto
  3. Alessandro Gimigliano
  4. Enrico Carlini
  5. M. Catalisano

In Computational Theory and Mathematics

  1. Alessandra Bernardi
  2. M. Catalisano
  3. A. Oneto
  4. Alessandro Gimigliano
  5. Enrico Carlini

In Geometry and Topology

  1. A. Oneto
  2. M. Catalisano
  3. Alessandro Gimigliano
  4. Alessandra Bernardi
  5. Enrico Carlini

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