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Mathematics of Computation

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The monic rank

Authors: Arthur Bik, Jan Draisma, Alessandro Oneto and Emanuele Ventura
Journal: Math. Comp. 89 (2020), 2481-2505
MSC (2010): Primary 15A21, 14R20, 13P10
Published electronically: February 20, 2020
MathSciNet review: 4109574
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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.

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Additional Information

Arthur Bik
Affiliation: Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland

Jan Draisma
Affiliation: Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland; and Eindhoven University of Technology, 5612 Eindhoven, Netherlands

Alessandro Oneto
Affiliation: Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

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

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.
Article copyright: © Copyright 2020 American Mathematical Society