The edge of the asymptotic spectrum of tensors

Abstract

Strassen founded the theory of the asymptotic spectrum of tensors to study the complexity of matrix multiplication. A central challenge in this theory is to explicitly construct new spectral points. In Crelle 1991, Strassen proposed the upper support functionals ${\zeta }^{\theta }$ as candidate spectral points, where $\theta$ ranges over a triangle $\Theta$. Recent progress, involving tools and ideas from quantum information theory (Christandl-Vrana-Zuiddam, STOC 2018, JAMS 2023) and convex optimization (Hirai, 2025), culminated in the proof that the upper support functionals are indeed spectral points over the complex numbers (Sakabe-Doğan-Walter, 2026). In this paper, we give an even clearer picture of the situation for support functionals when $\theta$ lies along the edges of the triangle. We show that not only are these functionals spectral points, but that they are uniquely determined as spectral points by their behavior on matrix multiplication tensors. As our methods are algebraic, as a corollary this gives the first explicit nontrivial spectral points over arbitrary fields. As part of our argument, we show a close connection between the edge support functionals and Harder-Narasimhan filtrations from quiver representation theory. We thus show, using recent work in algorithmic invariant theory, that these support functionals can be computed in deterministic polynomial time. Other ingredients of our proof include a new criterion for abstractly characterizing asymptotic tensor ranks by spectral points, and a characterization of the edge support functionals in terms of matrix multiplication capacity. As another application of these tools, we prove the existence of spectral points for higher-mode tensors beyond those currently known.

65 pages, abstract shortened for arXiv, to appear in FOCS 2026