Partially Symmetric Variants of Comon's Problem Via Simultaneous Rank

A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. We show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials.

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