Award Date

5-1-2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Arthur Baragar

Second Committee Member

Douglas Burke

Third Committee Member

Gennady Bachman

Fourth Committee Member

Kathy Robbins

Number of Pages

66

Abstract

The Markoff equation is x^2+y^2+z^2 = 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^2 + y^2 + z^2 = 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

{c_n} such that the tree generated from (1, c_n, c_n) 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.

Keywords

Chebyshev polynomials; Markoff equations; Markov processes; Markov spectrum; Unicity conjecture

Disciplines

Mathematics

Language

English


Included in

Mathematics Commons

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