Periodic harmonic functions on lattices and points count in positive characteristic

Mikhail Zaidenberg

Open Mathematics (2009)

  • Volume: 7, Issue: 3, page 365-381
  • ISSN: 2391-5455

Abstract

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This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

How to cite

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Mikhail Zaidenberg. "Periodic harmonic functions on lattices and points count in positive characteristic." Open Mathematics 7.3 (2009): 365-381. <http://eudml.org/doc/269497>.

@article{MikhailZaidenberg2009,
abstract = {This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.},
author = {Mikhail Zaidenberg},
journal = {Open Mathematics},
keywords = {Cellular automaton; Chebyshev-Dickson polynomial; Convolution operator; Lattice; Finite field; Discrete Fourier transform; Discrete harmonic function; Pluri-periodic function; cellular automaton; convolution operator; lattice; finite field; discrete Fourier transform; discrete harmonic function; pluri-periodic function},
language = {eng},
number = {3},
pages = {365-381},
title = {Periodic harmonic functions on lattices and points count in positive characteristic},
url = {http://eudml.org/doc/269497},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Mikhail Zaidenberg
TI - Periodic harmonic functions on lattices and points count in positive characteristic
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 365
EP - 381
AB - This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.
LA - eng
KW - Cellular automaton; Chebyshev-Dickson polynomial; Convolution operator; Lattice; Finite field; Discrete Fourier transform; Discrete harmonic function; Pluri-periodic function; cellular automaton; convolution operator; lattice; finite field; discrete Fourier transform; discrete harmonic function; pluri-periodic function
UR - http://eudml.org/doc/269497
ER -

References

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  1. [1] Amin A.T., Slater P.J., Zhang G.-H., Parity dimension for graphs-a linear algebraic approach, Linear and Multilinear Algebra, 2002, 50, 327–342 http://dx.doi.org/10.1080/0308108021000049293 Zbl1023.05095
  2. [2] Barua R., Ramakrishnan S., σ-game, σ+-game and two-dimensional additive cellular automata, Theoret. Comput. Sci., 1996, 154, 349–366 http://dx.doi.org/10.1016/0304-3975(95)00091-7 
  3. [3] Bhargava M., Zieve M.E., Factoring Dickson polynomials over finite fields, Finite Fields Appl., 1999, 5, 103–111 http://dx.doi.org/10.1006/ffta.1998.0221 
  4. [4] Bicknell M., A primer for the Fibonacci numbers VII, Fibonacci Quart., 1970, 8, 407–420 
  5. [5] Goldwasser J., Klostermeyer W., Ware H., Fibonacci Polynomials and Parity Domination in Grid Graphs, Graphs Combin., 2002, 18, 271–283 http://dx.doi.org/10.1007/s003730200020 Zbl0999.05079
  6. [6] Goldwasser J., Wang X., Wu Y., Does the lit-only restriction make any difference for the σ-game and σ+-game?, European J. Combin., 2009, 30, 774–787 http://dx.doi.org/10.1016/j.ejc.2008.09.020 Zbl1169.91011
  7. [7] Gravier S., Mhalla M., Tannier E., On a modular domination game, Theoret. Comput. Sci., 2003, 306, 291–303 http://dx.doi.org/10.1016/S0304-3975(03)00285-8 Zbl1060.68076
  8. [8] Heath-Brown D.R., Artin’s conjecture for primitive roots, Quart. J. Math., 1986, 37, 27–38 http://dx.doi.org/10.1093/qmath/37.1.27 Zbl0586.10025
  9. [9] Hoggatt V.E.Jr., Bicknell-Johnson M., Divisibility properties of polynomials in Pascal’s triangle, Fibonacci Quart., 1978, 16, 501–513 Zbl0388.10009
  10. [10] Hoggatt V.E.Jr., Long C.T., Divisibility properties of generalized Fibonacci polynomials, Fibonacci Quart., 1974, 12, 113–120 Zbl0284.10003
  11. [11] Humphreys J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1978 Zbl0447.17001
  12. [12] Hunziker M., Machiavelo A., Park J., Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids, Theoret. Comput. Sci., 2004, 320, 465–483 http://dx.doi.org/10.1016/j.tcs.2004.03.031 Zbl1068.68089
  13. [13] Jacob G., Reutenauer C., Sakarovitch J., On a divisibility property of Fibonacci polynomials, preprint available at http://en.scientificcommons.org/43936584 
  14. [14] Levy D., The irreducible factorization of Fibonacci polynomials over Q, Fibonacci Quart., 2001, 39, 309–319 Zbl1009.11012
  15. [15] Lidl R., Mullen G.L., Turnwald G., Dickson polynomials, Longman Scientific and Technical, Harlow, John Wiley and Sons, Inc., New York, 1993 
  16. [16] Martin O., Odlyzko A.M., Wolfram S., Algebraic properties of cellular automata, Comm. Math. Phys., 1984, 93, 219–258 http://dx.doi.org/10.1007/BF01223745 Zbl0564.68038
  17. [17] Moree P., Artin’s primitive root conjecture-a survey, preprint available at http://arxiv.org/abs/math/0412262 Zbl1271.11002
  18. [18] Ram Murty M., Artin’s conjecture for primitive roots, Math. Intelligencer, 1988, 10, 59–67 http://dx.doi.org/10.1007/BF03023749 Zbl0656.10044
  19. [19] Sarkar P., Barua R., Multidimensional σ-automata, π-polynomials and generalised S-matrices, Theoret. Comput. Sci., 1998, 197, 111–138 http://dx.doi.org/10.1016/S0304-3975(97)00160-6 Zbl0902.68125
  20. [20] Sutner K., σ-automata and Chebyshev-polynomials, Theoret. Comput. Sci., 2000, 230, 49–73 http://dx.doi.org/10.1016/S0304-3975(97)00242-9 
  21. [21] Webb W.A., Parberry E.A., Divisibility properties of Fibonacci polynomials, Fibonacci Quart., 1969, 7, 457–463 Zbl0211.37303
  22. [22] Zaidenberg M., Periodic binary harmonic functions on lattices, Adv. in Appl. Math., 2008, 40, 225–265 http://dx.doi.org/10.1016/j.aam.2007.01.004 
  23. [23] Zaidenberg M., Convolution equations on lattices: periodic solutions with values in a prime characteristic field, In: Kapranov M., Kolyada S., Manin Y.I., Moree P., Potyagailo L.A. (Eds.), Geometry and Dynamics of Groups and Spaces, In Memory of Alexander Reznikov, Progress in Mathematics 265, 719–740, Birkhäuser, 2008 

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