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Comon Conjecture

The Comon conjecture asserts that the tensor decomposition rank and symmetric tensor rank of a symmetric tensor are equal (Comon et al. 2008). Writing these ranks as r(T) and s(T), respectively, one always has r(T)<=s(T) because every symmetric decomposition is also an unrestricted tensor decomposition.

Shitov's (2018) proposed counterexample over the complex numbers contained an error in the lower bound on its symmetric tensor rank (Draisma 2024). Lovitz (2026) constructed a smaller symmetric tensor T_0 with rational entries of size 27×27×27 such that r(T_0)=55 and s(T_0)=56 over both the real and complex numbers, disproving the conjecture over either field. Li (2026) proved that the tensor direct sum of two copies has

(r(T_0 direct sum T_0),s(T_0 direct sum T_0))=(110,112),

while three copies have

(r(T_0 direct sum T_0 direct sum T_0),s(T_0 direct sum T_0 direct sum T_0))=(165,168).

These tensors have sizes 54×54×54 and 81×81×81, respectively. Consequently, the difference s(T)-r(T) can be 2 or 3, not only 1.

Lovitz (2026) reports using GPT-6 Astra throughout the project, including to obtain an initial proof that he subsequently verified and simplified. Li (2026) reports using ChatGPT throughout the mathematical development and manuscript preparation and takes responsibility for the paper.

See also

Symmetric Tensor, Symmetric Tensor Rank, Tensor Decomposition, Tensor Decomposition Rank, Tensor Direct Sum

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References

Comon, P.; Golub, G.; Lim, L.-H.; and Mourrain, B. "Symmetric Tensors and Symmetric Tensor Rank." SIAM J. Matrix Anal. Appl. 30, 1254-1279, 2008. https://doi.org/10.1137/060661569.Draisma, J. "Erratum: A Counterexample to Comon's Conjecture." SIAM J. Appl. Algebra Geom. 8, 225, 2024. https://doi.org/10.1137/23M1623781.Li, J. "The Comon Rank Gap of a Third-Order Symmetric Tensor Can Exceed One." 1 Oct 2026. https://arxiv.org/abs/2610.01660.Lovitz, B. "A 27×27×27 Counterexample to Comon's Conjecture." 23 Sep 2026. https://arxiv.org/abs/2609.28292.Shitov, Y. "A Counterexample to Comon's Conjecture." SIAM J. Appl. Algebra Geom. 2, 428-443, 2018. https://doi.org/10.1137/17M1131970.

Cite this as:

Weisstein, Eric W. "Comon Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComonConjecture.html

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