Sum of Squares and Non Sum of Squares Regimes for the Toeplitz Bottcher Wenzel Form
Abstract
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 squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every 2≤ N≤50, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.
Related
- Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form
- Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture
- Block Toeplitz Operators, Matrix Valued Truncated Toeplitz operators, and Conjugations
- A Non-Asymptotic Gaussian Likelihood Theory for Long-Memory Toeplitz Models
- Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
BibTeX
@misc{zhu2026squares,
title = {{Sum of Squares and Non Sum of Squares Regimes for the Toeplitz Bottcher Wenzel Form}},
author = {Wenqi Zhu and Ping Nie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08980},
eprint = {2610.08980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08980},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Sum of Squares and Non Sum of Squares Regimes for the Toeplitz Bottcher Wenzel Form
AU - Wenqi Zhu
AU - Ping Nie
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.08980
UR - https://arxiv.org/abs/2610.08980
AB - 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 squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every 2≤ N≤50, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.
ER -