Periodic binary harmonic functions on lattices

• Published in: Advances in applied mathematics Vol. 40; no. 2; pp. 225 - 265
• Main Author: Zaidenberg, Mikhail
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.02.2008; Elsevier
• Subjects: Algebra; Algebraic geometry; Binary function; Cellular automaton; Characteristic two; Chebyshev–Dickson polynomial; Combinatorics. Ordered structures; Discrete harmonic function; Elliptic curve; Exact sciences and technology; Fibonacci polynomial; Finite abelian group; General mathematics; General, history and biography; Graph Laplacian; Lattice; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Order, lattices, ordered algebraic structures; Pluri-periodic function; Sciences and techniques of general use; Spectrum of a graph; 11B39; Binary function; 37B15; Elliptic curve; Lattice; Characteristic two; 31C05; 11T06; Graph Laplacian; Cellular automaton; Chebyshev–Dickson polynomial; 43A99; Spectrum of a graph; Finite abelian group; Fibonacci polynomial; Discrete harmonic function; Pluri-periodic function; 11T99; Abelian group; Complex system; Periodic function; 43A99. Chebyshev-Dickson polynomial; Grid pattern; Automaton; Chebyshev polynomial; Harmonic function; Laplacian; Integer; Finite group; Applied mathematics
• ISSN: 0196-8858; 1090-2074
• DOI: 10.1016/j.aam.2007.01.004

A function on a (generally infinite) graph Γ with values in a field K of characteristic 2 will be called harmonic if its value at every vertex of Γ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions f : Z s → F 2 = GF ( 2 ) on integer lattices, and address the problem of describing the set of possible multi-periods n ¯ = ( n 1 , … , n s ) ∈ N s of such functions. This problem arises in the theory of cellular automata [O. Martin, A.M. Odlyzko, S. Wolfram, Algebraic properties of cellular automata, Comm. Math. Phys. 93 (1984) 219–258; K. Sutner, On σ-automata, Complex Systems 2 (1988) 1–28; K. Sutner, The σ-game and cellular automata, Amer. Math. Monthly 97 (1990) 24–34; J. Goldwasser, W. Klostermeyer, H. Ware, Fibonacci polynomials and parity domination in grid graphs, Graphs Combin. 18 (2002) 271–283]. It happens to be equivalent to determining, for a certain affine algebraic hypersurface V s in A F ¯ 2 s , the torsion multi-orders of the points on V s in the multiplicative group ( F ¯ 2 × ) 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.