# 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 $\bar n=(n_1,...,n_s)\in\N^s$ 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.
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Advances in Applied Mathematics, Elsevier, 2008, 40 (2), pp.225-265. 〈10.1016/j.aam.2007.01.004〉
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https://hal.archives-ouvertes.fr/hal-00425088
Contributeur : Mikhail Zaidenberg <>
Soumis le : mardi 20 octobre 2009 - 09:56:45
Dernière modification le : jeudi 11 janvier 2018 - 06:12:13

### Citation

Mikhail Zaidenberg. Periodic binary harmonic functions. Advances in Applied Mathematics, Elsevier, 2008, 40 (2), pp.225-265. 〈10.1016/j.aam.2007.01.004〉. 〈hal-00425088〉

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