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Towards a geometric approach to Strassen’s asymptotic rank conjecture

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Abstract

We make a first geometric study of three varieties in \(\mathbb {C}^m\otimes \mathbb {C}^m\otimes \mathbb {C}^m\) (for each m), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.

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Acknowledgements

We thank the anonymous referees for their valuable comments on an earlier version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Austin Conner, Joseph M. Landsberg & Yao Wang

  2. QMATH - Department Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen O., Denmark

    Fulvio Gesmundo

  3. Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Emanuele Ventura

Authors
  1. Austin Conner
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  2. Fulvio Gesmundo
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  3. Joseph M. Landsberg
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  4. Emanuele Ventura
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  5. Yao Wang
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Correspondence to Fulvio Gesmundo.

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Landsberg is supported by NSF Grants DMS-1405348 and AF-1814254 and by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund as well as a Simons Visiting Professor Grant supplied by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach. Gesmundo acknowledges financial support from the European Research Council (ERC Grant Agreement No. 337603), the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and the ICERM program in Nonlinear Algebra in Fall 2018 (NSF DMS-1439786). Ventura acknowledges financial support by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund.

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Conner, A., Gesmundo, F., Landsberg, J.M. et al. Towards a geometric approach to Strassen’s asymptotic rank conjecture. Collect. Math. 72, 63–86 (2021). https://doi.org/10.1007/s13348-020-00280-8

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  • DOI: https://doi.org/10.1007/s13348-020-00280-8

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