+publisher:"University of Colorado" +contributor:("Robert Tubbs")

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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 (g_i)_i_≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (f_i) _i_≥1 by the following recursion: f₁ = g₁, f_n = g_n ∘ f_n_–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 f_n(α) = [special characters omitted] is written in lowest terms, then for all but finitely many n > 0, the numerator, A_n, has a primitive divisor; that is, there is a prime p which divides A_n but does not divide A_i 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 f_n(x) = g_n( f_n_–1(x)) with g_i 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 g_i be a sequence of Chebyshev polynomials. Let f given by the following recurrence f₁(z) = g₁(z), and f_i = g_i( f_i_–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 f_n(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 (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

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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

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