Bik, Arthur; Draisma, Jan; Oneto, Alessandro; Ventura, Emanuele (2020). The monic rank. Mathematics of computation, 89(325), pp. 24812505. American Mathematical Society 10.1090/mcom/3512

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We introduce the monic rank of a vector relative to an affinehyperplane section of an irreducible Zariskiclosed 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 representationthis includes the wellknown cases of tensor rank and symmetric rankwe 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 and Ventura, Emanuele 
Subjects: 
500 Science > 510 Mathematics 
ISSN: 
00255718 
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 20152019 Polish MNiSW fund 
Language: 
English 
Submitter: 
Sebastiano Don 
Date Deposited: 
28 Jan 2021 18:04 
Last Modified: 
11 Mar 2021 20:15 
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 