Periodic binary harmonic functions

Abstract

A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called {\it harmonic} if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions $f: \Z^s\to\F_2=\GF(2)$ on integer lattices, and address the problem of describing the set of possible multi-periods $\bar{n}=(n_1,...,n_s)\in {\mathbb{N}}^s$ of such functions. Actually this problem arises in the theory of cellular automata. It occurs to be equivalent to determining, for a certain affine algebraic hypersurface $V_s$ in $\A_{\bar\F_2}^s$, the torsion multi-orders of the points on $V_s$ in the multiplicative group $(\bar\F_2^\times)^s$. In particular $V_2$ is an elliptic cubic curve. In this special case we provide a more thorough treatment. A major part of the paper is devoted to a survey of the subject.

36 pages, 3 figures