Eventually Stable Quadratic Polynomials over Q(i)
| dc.contributor.advisor | Washington, Lawrence C | en_US |
| dc.contributor.author | McDermott, Jermain | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2024-09-23T06:35:22Z | |
| dc.date.available | 2024-09-23T06:35:22Z | |
| dc.date.issued | 2024 | en_US |
| dc.description.abstract | Let $f$ be a polynomial or a rational function over a field $K$. Arithmetic dynamics studies the algebraic and number-theoretic properties of its iterates $f^n:=f \circ f \circ ... \circ f$.\\ A basic question is, if $f$ is a polynomial, are these iterates irreducible or not? We wish to know what can happen when considering iterates of a quadratic $f= x^2+r\in K[x]$. The most interesting case is when $r=\frac{1}{c}$, which we will focus on, and discuss criteria for irreducibility, i.e. \emph{stability} of all iterates. We also wish to prove that if 0 is not periodic under $f$, then the number of factors of $f^n$ is bounded by a constant independent of $n$, i.e. $f$ is \emph{eventually stable}. This thesis is an extension to $\Qi$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. This thesis involves a mixture of ideas from number theory and arithmetic geometry. We also show how eventual stability of iterates ties into the density of prime divisors of sequences. | en_US |
| dc.identifier | https://doi.org/10.13016/kmgc-iasa | |
| dc.identifier.uri | http://hdl.handle.net/1903/33486 | |
| dc.language.iso | en | en_US |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.title | Eventually Stable Quadratic Polynomials over Q(i) | en_US |
| dc.type | Dissertation | en_US |
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