Diophantine problems in polynomial theory

• Main Author: Lee, Paul David
• Language: English
• Published: University of British Columbia 2011
• Online Access: http://hdl.handle.net/2429/38252

Algebraic curves and surfaces are playing an increasing role in modern mathematics. From the well known applications to cryptography, to computer vision and manufacturing, studying these curves is a prevalent problem that is appearing more often. With the advancement of computers, dramatic progress has been made in all branches of algebraic computation. In particular, computer algebra software has made it much easier to find rational or integral points on algebraic curves. Computers have also made it easier to obtain rational parametrizations of certain curves and surfaces. Each algebraic curve has an associated genus, essentially a classification, that determines its topological structure. Advancements on methods and theory on curves of genus 0, 1 and 2 have been made in recent years. Curves of genus 0 are the only algebraic curves that you can obtain a rational parametrization for. Curves of genus 1 (also known as elliptic curves) have the property that their rational points have a group structure and thus one can call upon the massive field of group theory to help with their study. Curves of higher genus (such as genus 2) do not have the background and theory that genus 0 and 1 do but recent advancements in theory have rapidly expanded advancements on the topic. In this thesis, we will first outline some methods used to find rational and integral points on curves of genus 0, 1, and 2. We will then solve some new problems related to polynomial theory that require finding the solutions to systems of Diophantine equations. We are required to find rational or integral points on algebraic curves to garner the solutions to these systems.