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You searched for +publisher:"University of Illinois – Chicago" +contributor:("Cojocaru, Alina"). Showing records 1 – 2 of 2 total matches.

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University of Illinois – Chicago

1. Mullen, Cara. The Critical Orbit Structure of Quadratic Polynomials in Zp.

Degree: 2017, University of Illinois – Chicago

In this thesis, we develop a non-Archimedean analog to the Hubbard tree, a well-understood object from classical dynamics studied over the complex numbers. To that end, we explore the critical orbit structure of quadratic polynomials fc(z) = z2 + c with parameters c in the ring of p-adic rational integers, Zp. All such polynomials are post-critically bounded (PCB), and some are post-critically finite (PCF), which means that the forward orbit of the critical point, 0, is finite. If the orbit of 0 is finite, there exist minimal integers m and n such that fcm+n(0)=fcm(0), and we call (m,n) the critical orbit type of fc. All PCF polynomials f defined over the complex numbers have an associated Hubbard tree which illustrates the orbit type of the critical points, and the geometry of those orbits within the filled Julia set of f. In order to define a p-adic analog, we give a description of the exact critical orbit type for PCF polynomials fc defined over Zp, and then use the proximity of PCB parameters to PCF points in order to prove statements about the structure of the infinite critical orbit as visualized in the Zp tree. Advisors/Committee Members: DeMarco, Laura (advisor), Cojocaru, Alina (committee member), Hurder, Steve (committee member), Lindsey, Kathryn (committee member), Takloo-Bighash, Ramin (chair).

Subjects/Keywords: Arithmetic Dynamics; Number Theory

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

APA (6th Edition):

Mullen, C. (2017). The Critical Orbit Structure of Quadratic Polynomials in Zp. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/21816

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Mullen, Cara. “The Critical Orbit Structure of Quadratic Polynomials in Zp.” 2017. Thesis, University of Illinois – Chicago. Accessed July 03, 2020. http://hdl.handle.net/10027/21816.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Mullen, Cara. “The Critical Orbit Structure of Quadratic Polynomials in Zp.” 2017. Web. 03 Jul 2020.

Vancouver:

Mullen C. The Critical Orbit Structure of Quadratic Polynomials in Zp. [Internet] [Thesis]. University of Illinois – Chicago; 2017. [cited 2020 Jul 03]. Available from: http://hdl.handle.net/10027/21816.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Mullen C. The Critical Orbit Structure of Quadratic Polynomials in Zp. [Thesis]. University of Illinois – Chicago; 2017. Available from: http://hdl.handle.net/10027/21816

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Shulman, Andrew. Elementary divisors of reductions of generic Drinfeld modules.

Degree: 2012, University of Illinois – Chicago

Let q be a power of an odd prime, A := Fq[T], and k := Fq(T). Let ψ be a Drinfeld A- module over a fi nite extension K of k of rank r  ≥  2. Let \wp be a prime of K, of good reduction for ψ, \F\wp the residue fi eld at \wp, and consider the reduced Drinfeld A-module ψ \otimes \wp over \F\wp. The A-module action of ψ \otimes \wp on \F\wp, denote \F\wp, makes \F\wp isomorphic, as an A-module, to A / d1,\wp(ψ)A  ×  …  ×  A / dr,\wp(ψ) A for uniquely determined monic polynomials d1,\wp(ψ),…,dr,\wp(ψ), depending on ψ and \wp, such that d1,\wp(ψ) | … | dr,\wp(ψ). The elements d1,\wp(ψ),…,dr,\wp(ψ) are called the elementary divisors of ψ modulo \wp. In this thesis, we study the growth of the largest elementary divisor, dr,\wp(ψ), as the prime \wp varies, in analogy with a result by W. Duke pertaining to elliptic curves. We also consider the distribution of the smallest elementary divisor, d1,\wp(ψ), again as \wp varies, in analogy with work started by J.-P. Serre related to Lang and Trotter's elliptic curve formulation of Artin's primitive root conjecture. One of our main results is that for a density 1 of primes \wp of K, the infinity norm of dr,\wp(ψ) is as large as possible. More precisely, we show that for any function f de fined on the primes of K with values in A such that f grows very slowly as the degree of the primes in K increases to infinity, then almost all primes \wp of K satisfy |dr,\wp(ψ)|\ifnty > \frac{|\wp|}{|f(\wp)|. Advisors/Committee Members: Cojocaru, Alina Carmen (advisor), Coskun, Izzet (committee member), Popa, Mihnea (committee member), Talkoo-Bighash, Ramin (committee member), Zieve, Michael (committee member).

Subjects/Keywords: Drinfeld modules; elliptic curves; reduction; elementary divisors

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

APA (6th Edition):

Shulman, A. (2012). Elementary divisors of reductions of generic Drinfeld modules. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/8268

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Shulman, Andrew. “Elementary divisors of reductions of generic Drinfeld modules.” 2012. Thesis, University of Illinois – Chicago. Accessed July 03, 2020. http://hdl.handle.net/10027/8268.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Shulman, Andrew. “Elementary divisors of reductions of generic Drinfeld modules.” 2012. Web. 03 Jul 2020.

Vancouver:

Shulman A. Elementary divisors of reductions of generic Drinfeld modules. [Internet] [Thesis]. University of Illinois – Chicago; 2012. [cited 2020 Jul 03]. Available from: http://hdl.handle.net/10027/8268.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Shulman A. Elementary divisors of reductions of generic Drinfeld modules. [Thesis]. University of Illinois – Chicago; 2012. Available from: http://hdl.handle.net/10027/8268

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.