Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2018; 55(5): 1587-1598

Online first article March 15, 2018      Printed September 30, 2018

https://doi.org/10.4134/BKMS.b170933

Copyright © The Korean Mathematical Society.

Beyond the cactus rank of tensors

Edoardo Ballico

University of Trento

Abstract

We study additive decompositions (and generalized additive decompositions with a zero-dimensional scheme instead of a finite sum of rank $1$ tensors), which are not of minimal degree (for sums of rank $1$ tensors with more terms than the rank of the tensor, for a zero-dimensional scheme a degree higher than the cactus rank of the tensor). We prove their existence for all degrees higher than the rank of the tensor and, with strong assumptions, higher than the cactus rank of the tensor. Examples show that additional assumptions are needed to get the minimally spanning scheme of degree cactus $+1$.

Keywords: tensor rank, tensor decomposition, cactus rank, zero-dimensional scheme, Segre embedding

MSC numbers: 14N05, 15A69