We introduce the monic rank of a vector relative to an affinehyperplane 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 · e is the sum of d dth 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.

The monic rank / Bik, A.; Draisma, J.; Oneto, A.; Ventura, E.. - In: MATHEMATICS OF COMPUTATION. - ISSN 0025-5718. - 89:325(2020), pp. 2481-2505. [10.1090/mcom/3521]

The monic rank

Oneto A.;
2020

Abstract

We introduce the monic rank of a vector relative to an affinehyperplane 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 · e is the sum of d dth 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.
325
Bik, A.; Draisma, J.; Oneto, A.; Ventura, E.
The monic rank / Bik, A.; Draisma, J.; Oneto, A.; Ventura, E.. - In: MATHEMATICS OF COMPUTATION. - ISSN 0025-5718. - 89:325(2020), pp. 2481-2505. [10.1090/mcom/3521]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11572/299187
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