Convolution equations on lattices: periodic solutions with values in a prime characteristic field

TL;DR

This paper studies convolution equations on lattices over fields of positive characteristic, linking solutions to algebraic hypersurfaces and harmonic analysis, inspired by cellular automata theory.

Contribution

It introduces a harmonic analysis approach for convolution operators over prime characteristic fields, extending cellular automata theory to algebraic hypersurfaces.

Findings

Spectral problems correspond to counting points on algebraic hypersurfaces.

Solutions are characterized via torsion orders of points on hypersurfaces.

The approach generalizes cellular automata dynamics to algebraic geometry context.

Abstract

These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we deal with general convolution operators. We propose an approach via harmonic analysis which works over a field of positive characteristic. It occurs that a standard spectral problem for a convolution operator is equivalent to counting points on an associate algebraic hypersurface in a torus according to the torsion orders of their coordinates.