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

A comparison of different notions of ranks of symmetric tensors