Bik, Arthur; Draisma, Jan; Oneto, Alessandro; Ventura, Emanuele (2020). The monic rank. Mathematics of computation, 89(325), pp. 2481-2505. American Mathematical Society 10.1090/mcom/3512
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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.
Item Type: |
Journal Article (Original Article) |
---|---|
Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |
UniBE Contributor: |
Bik, Michel Arthur, Draisma, Jan, Ventura, Emanuele |
Subjects: |
500 Science > 510 Mathematics |
ISSN: |
0025-5718 |
Publisher: |
American Mathematical Society |
Funders: |
[UNSPECIFIED] NWO Vici grant entitled Stabilisation in Algebra and Geometry ; [UNSPECIFIED] Spanish Ministry of Economy and Competitiveness ; [UNSPECIFIED] Simons Foundation grant 346300 for IMPAN ; [UNSPECIFIED] matching 2015-2019 Polish MNiSW fund |
Language: |
English |
Submitter: |
Sebastiano Don |
Date Deposited: |
28 Jan 2021 18:04 |
Last Modified: |
05 Dec 2022 15:45 |
Publisher DOI: |
10.1090/mcom/3512 |
ArXiv ID: |
1901.11354 |
Uncontrolled Keywords: |
15A21, 14R20, 13P10 |
BORIS DOI: |
10.48350/151225 |
URI: |
https://boris.unibe.ch/id/eprint/151225 |