On minimal decompositions of low rank symmetric tensors

TL;DR

This paper develops an algebraic method using Apolarity Theory to construct minimal decompositions of low-rank symmetric tensors, providing a systematic approach for ranks up to 5, with implementation in Macaulay2.

Contribution

It introduces a novel algebraic procedure for minimal tensor decomposition based on apolarity and Hilbert functions, applicable to tensors of rank up to 5.

Findings

Procedure produces minimal apolar sets for tensors of rank ≤ 5

Implementation available in Macaulay2 software

Advances understanding of tensor decomposition structure

Abstract

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2.