The edge of the asymptotic spectrum of tensors

TL;DR

This paper advances the understanding of the asymptotic spectrum of tensors by characterizing support functionals along the edges of a key triangle, establishing their uniqueness, computability, and existence over arbitrary fields.

Contribution

It provides a clear algebraic characterization of edge support functionals, proves their uniqueness and computability, and establishes the existence of nontrivial spectral points over arbitrary fields.

Findings

Edge support functionals are spectral points uniquely determined by their behavior on matrix multiplication tensors.

Support functionals along the edges can be computed in deterministic polynomial time.

The existence of spectral points for higher-mode tensors is established.

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 2021) and convex optimization (Hirai, 2025), culminated in the proof that the upper support functionals are indeed spectral points over the complex numbers (Sakabe-Do\u{g}an-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 establishes for the first time the existence of 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.