Generalized Markoff Equations, Euclid Trees, and Chebyshev Polynomials

The Markoff equation is x+y+z = 3xyz, and all of the positive integer solutions of this equation occur on one tree generated from (1, 1, 1), which is called the Markoff tree. In this paper, we consider trees of solutions to equations of the form x + y + z = xyz + A. We say a tree of solutions satisfies the unicity condition if the maximum element of an ordered triple in the tree uniquely determines the other two. The unicity conjecture says that the Markoff tree satisifies the unicity condition. In this paper, we show that there exists a sequence of real numbers {cn} such that the tree generated from (1, cn, cn) satisfies the unicity condition for all n, and that these trees converge to the Markoff tree. We accomplish this by first recasting polynomial solutions as linear combinations of Chebyshev polynomials, and showing that these polynomials are distinct. Then we evaluate these polynomials at certain values and use a countability argument. We also obtain upper and lower bounds for these polynomial expressions.