The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions
Abstract
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 F2 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 ζXf(z)=1/(det(I−zcR(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 Bn. We also prove nonsingularity and justify the unique backward extension of the weight recurrence.
Related
- The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields
- Resolving a conjecture on quadratic APN functions and a new quadratic (n,n)-function associated to crooked functions
- Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
- The Complexity of KKT Points of Quadratic Programs with Polynomially-Bounded Coefficients
- Ergodic Linear-Quadratic Optimal Control with Stochastic Coefficients
BibTeX
@misc{cusick2026quadratic,
title = {{The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions}},
author = {Thomas W. Cusick},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01399},
eprint = {2609.01399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01399},
}
RIS
TY - UNPB
TI - The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions
AU - Thomas W. Cusick
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01399
UR - https://arxiv.org/abs/2609.01399
AB - 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)=1/(det(I−zcR(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.
ER -