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

Preview	Text 1901.11354.pdf - Submitted Version Available under License Publisher holds Copyright. Download (291kB) | Preview
	Text S0025-5718-2020-03512-1.pdf - Published Version Restricted to registered users only Available under License Publisher holds Copyright. Download (337kB) | Request a copy

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.

Interest & Impact

Downloads

87 since deposited on 28 Jan 2021
27 in the past 12 months

Citations

5 Citations in Web of Science ®
6 Citations in Scopus

Search

in Google Scholar™

Services

Actions (login required)

Edit item

Actions (login required)

Edit item