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University of Colorado

1. Wakefield, Nathan Paul. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.

Degree: PhD, Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/25

Let (<em>g_{i}</em>)<em>_{i}</em>_{ ≥1} be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>f_{i}</em>) <em>_{i}</em>_{ ≥1} by the following recursion: *f*_{1} = * g*_{1}, <em>f_{n}</em> = <em>g_{n}</em> ∘ <em> f_{n}</em>_{–1}, for *n* ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) : *n* ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if <em>f_{ n}</em>(α) = [special characters omitted] is written in lowest terms, then for all but finitely many * n* > 0, the numerator, <em>A_{n}</em>, has a primitive divisor; that is, there is a prime *p* which divides <em> A_{n}</em> but does not divide <em>A_{i}</em> for any *i* < *n*.
In addition to the main result, several of the tools developed to prove the main result may be of interest.
A key component of the main result was the development of a generalization of canonical height. Namely: If [*f*] is a set of rational maps, all commuting with a common function *f*, and f = [special characters omitted] is a generalized iteration of rational maps formed by <em>f_{ n}</em>(*x*) = <em>g_{n}</em>(<em> f_{n}</em>_{–1}(*x*)) with <em> g_{i}</em> coming from [*f*], then there is a unique canonical height funtion *ĥ*_{f} : *K* → [special characters omitted] which is identical to the canonical height function associated to * f.*
Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and <em>g_{i}</em> be a sequence of Chebyshev polynomials. Let *f* given by the following recurrence *f*_{ 1}(*z*) = *g*_{1}(*z*), and <em>f_{i}</em> = <em>g_{i}</em>(<em> f_{i}</em>_{–1}(*z*)) for * i* ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write <em>f_{ n}</em>(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.
*Advisors/Committee Members: Su-Ion Ih, Katherine Stange, Robert Tubbs, Eric Stade, Juan Restrepo.*

Subjects/Keywords: Arithmetic Dynamics; Chebyshev; Generalized Iteration; Primitive Divisors; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wakefield, N. P. (2013). Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/25

Chicago Manual of Style (16^{th} Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 18, 2021. https://scholar.colorado.edu/math_gradetds/25.

MLA Handbook (7^{th} Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Web. 18 Jan 2021.

Vancouver:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 18]. Available from: https://scholar.colorado.edu/math_gradetds/25.

Council of Science Editors:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/25

University of Colorado

2. Feaver, A.F. Amy. Euclid's Algorithm in Multiquadratic Fields.

Degree: PhD, Mathematics, 2014, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/30

In this thesis we find that all imaginary *n*-quadratic fields with *n*>3 have class number larger than 1 and therefore cannot be Euclidean. We also examine imaginary triquadratic fields, presenting a complete list of 17 imaginary triquadratic fields with class number 1, and classifing many of them according to whether or not they are norm-Euclidean. We find that at least three of these fields are norm-Euclidean, and at least five are not.
*Advisors/Committee Members: Katherine Stange, David Grant, Robert Tubbs, Eric Stade, Franck Vernerey.*

Subjects/Keywords: Class Number; Euclidean Rings; Number Fields; Mathematics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Feaver, A. F. A. (2014). Euclid's Algorithm in Multiquadratic Fields. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/30

Chicago Manual of Style (16^{th} Edition):

Feaver, A F Amy. “Euclid's Algorithm in Multiquadratic Fields.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 18, 2021. https://scholar.colorado.edu/math_gradetds/30.

MLA Handbook (7^{th} Edition):

Feaver, A F Amy. “Euclid's Algorithm in Multiquadratic Fields.” 2014. Web. 18 Jan 2021.

Vancouver:

Feaver AFA. Euclid's Algorithm in Multiquadratic Fields. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 18]. Available from: https://scholar.colorado.edu/math_gradetds/30.

Council of Science Editors:

Feaver AFA. Euclid's Algorithm in Multiquadratic Fields. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/math_gradetds/30