Detecting Direct Sums of Tensors and Their Limits
S. Canino, C. Flavi, J. Jelisiejew
TL;DR
The paper develops a unified, centroid-based framework to classify direct sums and their limits for tensors in Segre-Veronese formats, extending Mammana and related works to arbitrarily many factors. It shows that the centroid of a tensor encodes a rich structure: the presence of nilpotent subalgebras k[ε]/(ε^n) yields that the tensor is a limit of direct sums with n summands, and it provides explicit epsilon-type decompositions that connect to apolarity and differential operators. The approach yields practical criteria linking centroid data to secant-variety membership and border rank via minimal generators of apolar ideals, and it includes constructive deformation methods to realize limits. Collectively, these results generalize prior two-factor results, offer new tools for understanding tensor decompositions, and illuminate the geometry of gradient fibers and secant varieties in the Segre-Veronese setting.
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 Buczyńska-Buczyński-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.