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 An algebraic view of discrete geometry
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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  De Zeeuw, Frank 
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  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20111021 
Provider  Vancouver : University of British Columbia Library 
Rights  AttributionNonCommercialNoDerivatives 4.0 International 
DOI  10.14288/1.0072373 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Graduation Date  201111 
Campus  UBCV 
Scholarly Level  Graduate 
Rights URI  
Aggregated Source Repository  DSpace 
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AttributionNonCommercialNoDerivatives 4.0 International