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Journal Articles Linear Algebra and its Applications Year : 2014

A comparison of different notions of ranks of symmetric tensors

Alessandra Bernardi
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Jérôme Brachat
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Bernard Mourrain

Abstract

We introduce various notions of rank for a symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.
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Dates and versions

hal-00746967 , version 1 (30-10-2012)
hal-00746967 , version 2 (26-11-2012)

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Alessandra Bernardi, Jérôme Brachat, Bernard Mourrain. A comparison of different notions of ranks of symmetric tensors. Linear Algebra and its Applications, 2014, 460, pp.205-230. ⟨hal-00746967v2⟩
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