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arXiv · 2609.28292

A 27 x 27 x 27 counterexample to Comon's conjecture

Benjamin Lovitz
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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.

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Benjamin Lovitz. 2026-09-23. A 27 x 27 x 27 counterexample to Comon's conjecture. https://arxiv.org/abs/2609.28292

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