Generalized linear cellular automata in groups and difference Galois theory II

@misc{blazquezsanz2026generalized,
  title = {{Generalized linear cellular automata in groups and difference Galois theory II}},
  author = {David Blázquez-Sanz and Mario Alejandro Vergara Tapiero},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12280},
  eprint = {2610.12280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12280},
}

TY  - UNPB
TI  - Generalized linear cellular automata in groups and difference Galois theory II
AU  - David Blázquez-Sanz
AU  - Mario Alejandro Vergara Tapiero
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12280
UR  - https://arxiv.org/abs/2610.12280
AB  - In the first part of this series the σ-ring spanned by the periodic solutions of a generalized linear cellular automaton in a discrete group was shown to be Hopf--Galois, with pro-algebraic Galois group, while a Galois theory for the σ-ring spanned by the finite support solutions was left open, with parameterized difference Galois theory suggested as the tool. We apply that tool to automata over ℤ. The Fourier transform turns the automaton into a rank one equation σ(x̂)=α̂x̂ over a field on which δ=z d/dz acts, and we compute, and algorithmically decide, its parameterized Galois group: for genuine automata with symbols rational in the time variable it is either 𝔾_m(ℂ) or 𝔾_m, the dividing line being separability of the symbol, α̂(z,t)=γ(z)ψ(t). Finally, the parameterized group in the time direction decides holonomy in the space direction; as a corollary, the Stirling numbers of the first kind are not spatially holonomic.
ER  -