通过有限型移位和ζ函数研究二次易系数猜想
The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions
- University at Buffalo（布法罗大学）
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AI总结：
本文通过有限型移位和ζ函数，证明了二次易系数猜想，推导了易系数公式，并论证了权重递推的非奇异性与唯一反向延拓性。
AI中文摘要：
我们证明了由T. W. Cusick在《二次旋转对称函数权重的递推》（《离散应用数学》378卷，2026年，第93-101页）中提出的猜想1——二次易系数猜想。对于任意二次单项式旋转对称布尔函数的有限和，我们确定了规则矩阵的递推部分为带符号二元德布鲁因转移矩阵B。随后，我们在Chirvasitu与Cusick的符号动力学构造中，给出了与该布尔函数相关的有限型移位的一步表示。在辅助F₂坐标下的傅里叶变换将该移位的邻接矩阵分解为无符号德布鲁因块和带符号块B，因此该移位的动力学ζ函数为ζ_{X_f}(z)=1/det(I-zℛ(f))，其中ℛ(f)为规则矩阵。该等式将符号动力学提供的特征值与规则矩阵特征多项式的根按代数重数对应起来，所需的易系数公式由Bⁿ的迹推导得出。我们还证明了该矩阵的非奇异性，并为权重递推的唯一反向延拓提供了依据。
英文摘要：
We prove the Quadratic Easy Coefficients Conjecture stated as Conjecture 1 in T. W. Cusick, \emph{Recursions for quadratic rotation symmetric functions weights}, Discrete Applied Mathematics 378 (2026), 93--101. For an arbitrary finite sum of quadratic monomial rotation symmetric Boolean functions, we identify the recurrent part of the rules matrix with a signed binary de Bruijn transfer matrix $B$. We then give a one-step presentation of the finite-type shift associated with the Boolean function in the symbolic-dynamics construction of Chirvasitu and Cusick. Fourier transformation in an auxiliary $\F_2$ coordinate decomposes the adjacency matrix of this shift into an unsigned de Bruijn block and the signed block $B$. Consequently the dynamical zeta function is \[ ζ_{X_f}(z)=\frac{1}{\det(I-z\cR(f))}, \] where $\cR(f)$ is the rules matrix. This equality identifies, with their algebraic multiplicities, the characteristic values, supplied by symbolic dynamics, with the roots of the characteristic polynomial of the rules matrix. The desired easy coefficients formula follows from the trace of $B^n$. We also prove nonsingularity and justify the unique backward extension of the weight recurrence.
