Partially symmetric variants of Comon's problem via simultaneous rank

Abstract

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. By exploiting algebraic tools such as apolarity theory, 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. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials.

arXiv comment28 pp

References in corpus (14)

• Ranks of tensors and a generalization of secant varieties
• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• On the rank of a binary form
• On the cactus rank of cubics forms
• General Tensor Decomposition, Moment Matrices and Applications
• Waring decompositions of monomials
• Multigraded regularity: syzygies and fat points
• Tensor ranks on tangent developable of Segre varieties
• Waring loci and the Strassen conjecture
• Geometric lower bounds for generalized ranks
• On the number of Waring decompositions for a generic polynomial vector
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• On polynomials with given Hilbert function and applications
• Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."

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• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• A Gap in the Subrank of Tensors
• Lower bounds on the rank and symmetric rank of real tensors
• Border rank non-additivity for higher order tensors
• Symmetric rank of some reducible forms