Nonsingularity of some classes of matrices, and minimal solutions of Silverman's game on discrete sets

Abstract

We establish the irreducibility of each game in four infinite three-parameter families of even order Silverman games, and the major step in doing so is to prove that certain matrices A, related in a simple way to the payoff matrices, are nonsingular for all relevant values of the parameters. This nonsingularity is established by, in effect, producing a matrix D such that AD is known to be nonsingular. The elements of D are polynomials from six interrelated sequences of polynomials closely related to the Chebyshev polynomials of the second kind. Each of these sequences satisfies a second order recursion, and consequently has many Fibonacci-like properties, which play an essential role in proving that the product AD is what we claim it is. The matrices D were found experimentally, by discovering patterns in low order cases worked out with the help of some computer algebra systems. The corresponding results for four families of odd order games were reported in an earlier paper.