# The monic rank

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 and Ventura, Emanuele

#### Subjects:

500 Science > 510 Mathematics

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

English

Sebastiano Don

#### Date Deposited:

28 Jan 2021 18:04

11 Mar 2021 20:15

#### Publisher DOI:

10.1090/mcom/3512

1901.11354

#### Uncontrolled Keywords:

15A21, 14R20, 13P10

10.48350/151225

#### URI:

https://boris.unibe.ch/id/eprint/151225