Periodic binary harmonic functions on lattices

Abstract

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-periodsn = (n 1 , . . . , n s ) ∈ N s of such functions. Actually this problem arises in the theory of cellular automata [MOW, Su1, Su4, GKW]. It occurs to be equivalent to determining, for a certain affine algebraic hypersurface V s in A s F2 , 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.In mathematics, our role is more that of servant than master.

A singular quartic curve over a finite field and the Trisentis game

“…[4,12]). In [4], we observed interesting patterns in the table of the dimension of the kernel of the Laplacian for Lights Out puzzle, and proved one of them by using the multiplication by 2 map on this elliptic curve.…”

“…have been extensively investigated. See, for example, [1,3,4,6,[9][10][11][12][13][14]. For these graphs, the σ + -game is mathematically much deeper than the σ -game.…”

“…We follow the notation in [8], which is slightly different from that in [12]. We believe that there is no fear of confusion between the cycle graph C n and the Chebyshev polynomial C n (x).…”