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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The monic rank
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by Arthur Bik, Jan Draisma, Alessandro Oneto and Emanuele Ventura HTML | PDF
Math. Comp. 89 (2020), 2481-2505 Request permission

Abstract:

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation—this includes the well-known cases of tensor rank and symmetric rank—we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.
References
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Additional Information
  • Arthur Bik
  • Affiliation: Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland
  • MR Author ID: 1297289
  • Email: arthur.bik@math.unibe.ch
  • Jan Draisma
  • Affiliation: Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland; and Eindhoven University of Technology, 5612 Eindhoven, Netherlands
  • MR Author ID: 683807
  • ORCID: 0000-0001-7248-8250
  • Email: jan.draisma@math.unibe.ch
  • Alessandro Oneto
  • Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
  • MR Author ID: 1087088
  • ORCID: 0000-0002-8142-6382
  • Email: aless.oneto@gmail.com, alessandro.oneto@ovgu.de
  • Emanuele Ventura
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Address at time of publication: Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland
  • Email: eventura@math.tamu.edu, emanueleventura.sw@gmail.com
  • Received by editor(s): January 31, 2019
  • Received by editor(s) in revised form: November 11, 2019
  • Published electronically: February 20, 2020
  • Additional Notes: The first author was supported by the second author’s Vici grant.
    The second author was partially supported by the NWO Vici grant entitled Stabilisation in Algebra and Geometry.
    The third author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R$\&$D (MDM-2014-0445).
    The fourth author acknowledges financial support by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 2481-2505
  • MSC (2010): Primary 15A21, 14R20, 13P10
  • DOI: https://doi.org/10.1090/mcom/3512
  • MathSciNet review: 4109574