OP-00248 Solution claimed

A sum-of-squares representation for the Toeplitz commutator form

For $n \ge 2$ let $x_{-(n-1)}, \dots, x_{n-1}, y_{-(n-1)}, \dots, y_{n-1}$ be independent real variables, let $X = (x_{i-j})_{i,j=1}^n$ and $Y = (y_{i-j})_{i,j=1}^n$ be the corresponding real Toeplitz matrices, and let $$F_n = 2\|X\|_F^2\|Y\|_F^2 - 2\bigl(\operatorname{tr}(X^{\mathsf T}Y)\bigr)^2 - \|XY - YX\|_F^2,$$ a homogeneous quartic polynomial in these $4n - 2$ variables, where $\|\cdot\|_F$ is the Frobenius norm. By the Böttcher–Wenzel inequality, $F_n \ge 0$ everywhere. For general real $n \times n$ matrices $P, Q$, László showed that the least $\gamma$ for which $(2 + \gamma)\bigl(\|P\|_F^2\|Q\|_F^2 - (\operatorname{tr}(P^{\mathsf T}Q))^2\bigr) - \|PQ - QP\|_F^2$ is a sum of squares of polynomials is $(n - 2)/2$; for Toeplitz $X, Y$ he showed that $F_n$ is a sum of squares for $n \le 7$.

Problem. Is it true that for every $n \ge 2$ there are an integer $N_n \ge 0$ and homogeneous quadratic polynomials $q_{n,1}, \dots, q_{n,N_n}$ with real coefficients in these $4n - 2$ variables such that $$F_n = \sum_{j=1}^{N_n} q_{n,j}^2$$ as an identity of polynomials?

Source: László (2012), Conjecture 15; Conjecture 4.1 in Ge, Li, Tang and Zhou's survey (2024; Conjecture 4.3 in arXiv:2402.01085), which credits it to Lu and Wenzel (2017).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: FA — Functional Analysis
Secondary: NA, OC
Keywords: Böttcher–Wenzel inequality, OpenProblemsInNLA, sums of squares, Toeplitz matrices
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{laszlo2012,
  author        = {L{\'a}szl{\'o}, Lajos},
  title         = {Sum of squares representation for the {B}{\"o}ttcher--{W}enzel biquadratic form},
  journal       = {Acta Univ. Sapientiae Inform.},
  volume        = {4},
  number        = {1},
  year          = {2012},
  pages         = {17--32},
  eprint        = {1207.6372},
  archivePrefix = {arXiv},
}
@article{ge2024,
  author        = {Ge, Jianquan and Li, Fagui and Tang, Zizhou and Zhou, Yi},
  title         = {A survey on the {DDVV}-type inequalities},
  journal       = {Adv. Math. (China)},
  volume        = {53},
  number        = {3},
  year          = {2024},
  pages         = {449--467},
  doi           = {10.11845/sxjz.2024001a},
  eprint        = {2402.01085},
  archivePrefix = {arXiv},
}
@incollection{lu2017,
  author    = {Lu, Zhiqin and Wenzel, David},
  title     = {Commutator estimates comprising the {F}robenius norm -- looking back and forth},
  booktitle = {Large Truncated {T}oeplitz Matrices, {T}oeplitz Operators, and Related Topics},
  editor    = {Bini, Dario A. and Ehrhardt, Torsten and Karlovich, Alexei Yu. and Spitkovsky, Ilya},
  series    = {Oper. Theory Adv. Appl.},
  volume    = {259},
  publisher = {Birkh{\"a}user/Springer},
  address   = {Cham},
  year      = {2017},
  pages     = {533--559},
  doi       = {10.1007/978-3-319-49182-0_22},
}
@article{bottcher2008,
  author  = {B{\"o}ttcher, Albrecht and Wenzel, David},
  title   = {The {F}robenius norm and the commutator},
  journal = {Linear Algebra Appl.},
  volume  = {429},
  number  = {8-9},
  year    = {2008},
  pages   = {1864--1885},
  doi     = {10.1016/j.laa.2008.05.020},
}
@misc{openproblemsinnla-mi15,
  author       = {Townsend, Alex},
  title        = {{MI-15}: A sum-of-squares representation for the Toeplitz commutator form},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-15/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Sum of Squares and Non Sum of Squares Regimes for the Toeplitz Bottcher Wenzel Form v1 arXiv ↗

Wenqi Zhu; Ping Nie

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the Böttcher--Wenzel form (BW form) is a nonnegative quartic measuring the gap in the corresponding commutator inequality. László conjectured that, when both matrices are Toeplitz, this quartic is a sum of …

Disproof · not verified · fidelity not established · posted on arXiv 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00248, "A sum-of-squares representation for the Toeplitz commutator form", clopen.solutions.