  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.
