发表机构
University of Oxford; University of Waterloo（牛津大学; 滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否证了 László 关于 Toeplitz 矩阵 Böttcher–Wenzel 形式为平方和的猜想，给出非平方和的显式反例族，并证明对称或反对称情形及阶数≤50 时仍为平方和。
AI 中文摘要
Toeplitz 矩阵是平移不变性最简单的结构化模型之一，自然出现在卷积、平稳协方差模型以及平移不变算子的离散化中。对于一对实矩阵，Böttcher–Wenzel 形式（BW 形式）是一个非负四次式，度量相应交换子不等式中的差距。László 猜想，当两个矩阵均为 Toeplitz 矩阵时，该四次式对每个矩阵阶数都是平方和。我们证明该猜想是错误的。更精确地，我们确定了一个显式的角支撑 Toeplitz 对族，仅保留两侧最外层的对角线，对于该族，在所有足够大的阶数下，相关的 BW 形式都不是平方和。证明将 Gram 表示的完整族简化为一个有限复数检验：其 Gram 不变部分发展出一个严格负的极限方向，而剩余的 Gram 相关贡献消失。我们还给出了矩阵阶数的显式充分阈值。该否定结果由两个正区域补充。如果任一 Toeplitz 因子是对称或反对称的，则相应的 BW 形式对每个阶数都是 SoS，而另一个因子可以是任意的实 Toeplitz 矩阵。对于无限制的 Toeplitz 对，我们进一步证明了对于每个 $2\le N\le50$ 的 SoS 可表示性，并通过 Lean 4 形式化验证至阶数 20。
英文摘要
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 $2\le N\le50$, with formal Lean 4 verification through order 20.
