The monic rank

Bik, Arthur; Draisma, Jan; Oneto, Alessandro; Ventura, Emanuele

doi:10.1090/mcom/3512

The monic rank
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by Arthur Bik, Jan Draisma, Alessandro Oneto and Emanuele Ventura HTML | PDF

Math. Comp. 89 (2020), 2481-2505 Request permission

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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