Periodic harmonic functions on lattices and points count in positive characteristic

Mikhail Zaidenberg (1)

1 IF - Institut Fourier

This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game ``Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.

Mots clés: Cellular automaton; Chebyshev-Dickson polynomial; convolution operator; lattice; finite field; discrete Fourier transform; discrete harmonic function; pluri-periodic function.; pluri-periodic function

Domaines: Géométrie algébrique [math.AG]; Théorie spectrale [math.SP]

Mikhail Zaidenberg. Periodic harmonic functions on lattices and points count in positive characteristic. Central European Journal of Mathematics, 2009, 7 (3), pp.365-381. ⟨10.2478/s11533-009-0029-0⟩. ⟨hal-00154845v2⟩