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

Abstract

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.