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On Comon’s and Strassen’s Conjectures

Author

Listed:
  • Alex Casarotti

    (Department of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, Italy)

  • Alex Massarenti

    (Department of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, Italy
    Institute of Mathematics and Statistics, Universidade Federal Fluminense, Niterói 24210-200, Brazil)

  • Massimiliano Mella

    (Department of Mathematics and Informatics, University of Ferrara, 44121 Ferrara, Italy)

Abstract

Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

Suggested Citation

  • Alex Casarotti & Alex Massarenti & Massimiliano Mella, 2018. "On Comon’s and Strassen’s Conjectures," Mathematics, MDPI, vol. 6(11), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:11:p:217-:d:178224
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