Detecting Direct Sums of Tensors and Their Limits

TL;DR

This paper extends the classification of limits of direct sums of tensors to multiple factors and general formats, providing a unified and characterizable framework using centroid theory, avoiding complex Betti number arguments.

Contribution

It generalizes Mammana's classification to more factors and formats, unifying previous results with a new centroid-based approach.

Findings

Limits of tensor direct sums can be characterized in general settings.

The approach avoids complex Betti number arguments.

Results unify and extend previous classifications.

Abstract

We generalize Mammana's classification of limits of direct sums to more than two factors. We also extend it from polynomials to arbitrary Segre-Veronese format, generalising and unifying results of Buczy\'nska-Buczy\'nski-Kleppe-Teitler, Hwang, Wang, and Wilson. Remarkably, in such much more general setup it is still possible to characterise the possible limits. Our proofs are direct and based on the theory of centroids, in particular avoiding the delicate Betti number arguments.