Partially symmetric variants of Comon's problem via simultaneous rank

TL;DR

This paper investigates how different notions of rank for symmetric tensors relate when considering partial symmetries, using algebraic tools to establish equalities among these ranks in various special cases.

Contribution

It introduces a method leveraging apolarity theory to analyze simultaneous symmetric ranks, clarifying the impact of tensor symmetries on rank equivalences.

Findings

Equalities among different partially symmetric ranks established

Application to binary, ternary, and quaternary cubics, monomials, and elementary symmetric polynomials

Enhanced understanding of symmetry effects on tensor 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.