## The monic rank

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Arthur Bik, Jan Draisma, Alessandro Oneto and Emanuele Ventura
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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
- 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