Alessandra Bernardi, Jérôme Brachat, Bernard Mourrain
Published 2012-10-30, updated 2012-11-27Version 2
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.
On the ranks and border ranks of symmetric tensors
On tensors of border rank $l$ in $\C^{m\times n\times l}$
The product of the eigenvalues of a symmetric tensor
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