An algebraic view of discrete geometry

• Main Author: De Zeeuw, Frank
• Language: English
• Published: University of British Columbia 2011
• Online Access: http://hdl.handle.net/2429/38158

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 two-dimensional 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.