Beyond the cactus rank of tensors
Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1587-1598
Published online September 30, 2018
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
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