Explicit 27 by 27 by 27 tensor disproves Comons conjecture
A 27 x 27 x 27 counterexample to Comon's conjecture
Data Structures and Algorithms
Summary
Comon's conjecture suggested that certain ways of measuring complexity in symmetric tensors would always match up. This paper finds a small, explicit example where those measurements differ, contradicting the conjecture. The authors build on previous theoretical ideas to prove a specific special case that leads to this counterexample. This result clarifies how symmetric tensors can behave in ways previously thought impossible.
What this means in practice
- •For mathematical software developers: Refine tensor decomposition algorithms by incorporating cases where symmetric rank differs from rank, improving tool accuracy for symmetric tensor analyses.
- •For quantum computing engineers: Adjust quantum state representations that rely on tensor decompositions to accommodate distinctions between symmetric and general tensor ranks highlighted by this result.
A theory result. No direct application yet.
Authors
Benjamin Lovitz
Abstract
We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.