Improving regulator verification and compact representations in real quadratic fields
Date
2013-01-08
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Abstract
The study of Diophantine equations, and in particular the erroneously-named Pell equation, has a long and intriguing history. In this work, we investigate solutions to the Pell equation and to a closely related quantity called the fundamental unit. Though it is somewhat simple to show these equations have non-trivial solutions, finding unconditionally correct solutions and being able to express them is extremely challenging, both from a theoretical and a computational perspective. We develop improvements to the algorithm presented by de Haan, Jacobson, and Williams [21, 22] which unconditionally verifies the regulator of a real quadratic number field and refinements to the concept of a compact representation of a quadratic integer, originally given by Buchmann, Thiel, and Williams [12]. In addition, we consider the well-known applications of this theory to principal ideal testing, finding integer points on elliptic curves, finding solutions to Diophantine equations, and some particular cryptographic applications.
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Mathematics, Mathematics
Citation
Silvester, A. (2013). Improving regulator verification and compact representations in real quadratic fields (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/26493