Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids

Hunziker, Markus; Machiavelo, António; Park, Jihun

doi:10.1016/j.tcs.2004.03.031

Cited by 20 publications

(10 citation statements)

References 11 publications

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“…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.…”

Section: Examplementioning

confidence: 99%

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

Yamagishi

2011

Finite Fields and Their Applications

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Communicated by Simeon Ball MSC: 05C50 11G20 14G15 31C20 91A46 Keywords: σ -game σ + -game Graph Laplacian Lights Out Strong product Kronecker product Chebyshev polynomialsThe Trisentis game consists of a rectangular array of lights each of which also functions as a toggle switch for its (up to eight) neighboring lights. The lights are OFF at the start, and the object is to turn them all ON. We give explicit formulas for the dimension of the kernel of the Laplacian associated to this game as well as some variants, in some cases, by counting rational points of the singular quartic curve (x + x −1 + 1)( y + y −1 + 1) = 1 over finite fields. As a corollary, we have an affirmative answer to a question of Clausing whether the n × n Trisentis game has a unique solution if n = 2 · 4 k or n = 2 · 4 k − 2.

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“…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.…”

Section: Examplementioning

confidence: 99%

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

Yamagishi

2011

Finite Fields and Their Applications

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“…[Bi,HBJ,HMP,§2] The normalized Chebyshev polynomials F n , G n and the Fibonacci polynomials f n acquire the following properties:…”

Section: Harmonic Functions On Treesmentioning

confidence: 99%

Periodic harmonic functions on lattices and points count in positive characteristic

Zaidenberg¹

2009

Open Mathematics

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“…For the proof of (a), (b), (d), (f) see [HMP],4.6,4.4,4.3,4.7 and 5.4, respectively. The proof of (e) in [HMP] is based on (d) and on a result of Heath-Brown [HB] on Artin's conjecture of primitive roots.…”

Section: Whereasmentioning

confidence: 99%

Periodic binary harmonic functions on lattices

Zaidenberg¹

2008

Advances in Applied Mathematics

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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.

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Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids

Cited by 20 publications

References 11 publications

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

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

Periodic harmonic functions on lattices and points count in positive characteristic

Periodic binary harmonic functions on lattices

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