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On real typical ranks

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Abstract

We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of n-variate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.

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Acknowledgements

We would like to thank F. O. Schreyer for useful talks about the paper [16]. We thank the Simons Institute for the Theory of Computing in Berkeley, CA for their generous support while in residence during the program on Algorithms and Complexity in Algebraic Geometry. The second author was partially supported by the Sloan Research Fellowship and NSF CAREER award DMS-1352073. A. Bernardi and G. Ottaviani are members of GNSAGA-INDAM.

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Authors and Affiliations

  1. Universita degli Studi di Trento Povo, Trento, Italy

    Alessandra Bernardi

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Grigoriy Blekherman

  3. Università di Firenze, Firenze, Italy

    Giorgio Ottaviani

Authors
  1. Alessandra Bernardi
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  2. Grigoriy Blekherman
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  3. Giorgio Ottaviani
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Correspondence to Alessandra Bernardi.

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Bernardi, A., Blekherman, G. & Ottaviani, G. On real typical ranks. Boll Unione Mat Ital 11, 293–307 (2018). https://doi.org/10.1007/s40574-017-0134-0

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