 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 An algebraic view of discrete geometry
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
An algebraic view of discrete geometry De Zeeuw, Frank
Abstract
This thesis includes three papers and one expository chapter as background for one of the papers. These papers have in common that they combine algebra with discrete geometry, mostly by using algebraic tools to prove statements from discrete geometry. Algebraic curves and number theory also recur throughout the proofs and results. In Chapter 1, we will detail these common threads. In Chapter 2, we prove that an infinite set of points in R² such that all pairwise distances are rational cannot be contained in an algebraic curve, except if that curve is a line or a circle, in which case at most 4 respectively 3 points of the set can be outside the line or circle. In the proof we use the classification of curves by their genus, and Faltings' Theorem. In Chapter 3, we informally present an elementary method for computing the genus of a planar algebraic curve, illustrating some of the techniques in Chapter 2. In Chapter 4, we prove a bound on the number of unit distances that can occur between points of a finite set in R², under the restriction that the line segments corresponding to these distances make a rational angle with the horizontal axis. In the proof we use graph theory and an algebraic theorem of Mann. In Chapter 5, we give an upper bound on the length of a simultaneous arithmetic progression (a twodimensional generalization of an arithmetic progression) on an elliptic curve, as well as for more general curves. We give a simple proof using a theorem of Jarnik, and another proof using the Crossing Inequality and some bounds from elementary algebraic geometry, which gives better explicit bounds.
Item Metadata
Title 
An algebraic view of discrete geometry

Creator  
Publisher 
University of British Columbia

Date Issued 
2011

Description 
This thesis includes three papers and one expository chapter as background
for one of the papers. These papers have in common that they combine
algebra with discrete geometry, mostly by using algebraic tools to prove
statements from discrete geometry. Algebraic curves and number theory
also recur throughout the proofs and results. In Chapter 1, we will detail
these common threads.
In Chapter 2, we prove that an infinite set of points in R² such that all
pairwise distances are rational cannot be contained in an algebraic curve,
except if that curve is a line or a circle, in which case at most 4 respectively 3
points of the set can be outside the line or circle. In the proof we use the
classification of curves by their genus, and Faltings' Theorem.
In Chapter 3, we informally present an elementary method for computing
the genus of a planar algebraic curve, illustrating some of the techniques in
Chapter 2.
In Chapter 4, we prove a bound on the number of unit distances that can
occur between points of a finite set in R², under the restriction that the line
segments corresponding to these distances make a rational angle with the
horizontal axis. In the proof we use graph theory and an algebraic theorem
of Mann.
In Chapter 5, we give an upper bound on the length of a simultaneous
arithmetic progression (a twodimensional generalization of an arithmetic
progression) on an elliptic curve, as well as for more general curves. We
give a simple proof using a theorem of Jarnik, and another proof using the
Crossing Inequality and some bounds from elementary algebraic geometry,
which gives better explicit bounds.

Genre  
Type  
Language 
eng

Date Available 
20111021

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0072373

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201111

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International