Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

Abstract

We show that if a homogeneous polynomial f in n variables has Waring rank \(n+1\), then the corresponding projective hypersurface \(f=0\) has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of f. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.

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Correspondence to Alexandru Dimca.

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\(^1\) This work has been partially supported by the French government, through the \(\mathrm UCA^{\mathrm{JEDI}}\) Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 and by the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III.

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Dimca, A., Sticlaru, G. Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces. Mediterr. J. Math. 17, 173 (2020). https://doi.org/10.1007/s00009-020-01609-0

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Keywords

  • Waring decomposition
  • Waring rank
  • Projective hypersurface
  • Isolated singularity
  • Hyperplane arrangement

Mathematics Subject Classification

  • Primary 14J70
  • Secondary 14B05
  • 32S05
  • 32S22