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Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 3, pp 293–307 | Cite as

On real typical ranks

  • Alessandra Bernardi
  • Grigoriy Blekherman
  • Giorgio Ottaviani
Article

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.

Mathematics Subject Classification

15A21 15A69 15A72 14P10 

Notes

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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Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Universita degli Studi di Trento PovoTrentoItaly
  2. 2.Georgia Institute of TechnologyAtlantaUSA
  3. 3.Università di FirenzeFirenzeItaly

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