Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2021; 58(1): 253-267

Online first article December 28, 2020      Printed January 31, 2021

https://doi.org/10.4134/BKMS.b200248

Copyright © The Korean Mathematical Society.

Linearly dependent and concise subsets of a Segre variety depending on $k$ factors

Edoardo Ballico

Via Sommarive 14

Abstract

We study linearly dependent subsets with prescribed cardinality $s$ of a multiprojective space. If the set $S$ is a circuit, there is an upper bound on the number of factors of the minimal multiprojective space containing $S$. B. Lovitz gave a sharp upper bound for this number. If $S$ has higher dependency, this may be not true without strong assumptions (and we give examples and suitable assumptions). We describe the dependent subsets $S$ with $\#S=6$.

Keywords: Segre varieties, tensor rank, tensor decomposition

MSC numbers: Primary 14N07, 14N05, 12E99, 12F99

Supported by: The author was partially supported by MIUR and GNSAGA of INdAM (Italy)

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