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

Authors:
Masakazu Yamagishi
Affiliations:
Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan
Venue:
Finite Fields and Their Applications
Year:
2011

Citing 6
Cited 0

The &sgr;-game and cellular automata

American Mathematical Monthly
σ-game, σ+-game and two-dimensional additive cellular automata

Theoretical Computer Science
Multidimensional &sgr;-automata, &pgr;-polynomials and generalised S-matrices

Theoretical Computer Science
&sgr;-Automata and Chebyshev-polynomials

Theoretical Computer Science
On a modular domination game

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

Theoretical Computer Science

Quantified Score

Hi-index	0.00

Visualization

Abstract

The 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 nxn Trisentis game has a unique solution if n=2@?4^k or n=2@?4^k-2.