Construct a Galois theory for finite-support solutions

Construct a Galois theory for the sigma-ring spanned by finite-support solutions of generalized linear cellular automata in discrete groups, extending beyond the periodic-solution case and determining the role of parameterized difference Galois theory.

Background

Generalized linear cellular automata give rise to several natural sigma-rings of solutions, including the ring spanned by periodic solutions and the ring spanned by finite-support solutions. The periodic case had already been treated using Picard–Vessiot–Franke theory, with a Hopf–Galois structure and a pro-algebraic Galois group established.

The paper studies the case of the group Z by applying parameterized difference Galois theory to the Fourier-transformed rank-one equation. However, the authors explicitly distinguish this computation from constructing a Galois theory for the finite-support solution ring itself. Thus, a complete Galois-theoretic description of that ring remains unresolved.

References

In the first part of this series the $\sigma$-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 $\sigma$-ring spanned by the finite support solutions was left open, with parameterized difference Galois theory suggested as the tool.

— Generalized linear cellular automata in groups and difference Galois theory II  (2610.12280 - Blázquez-Sanz et al., 8 Oct 2026) in Abstract; see also Section 1, Introduction