Minimum Euclidean Function Over the Eisenstein Integers

Abstract

There are many ways of computing distance in the real world. For instance, the distance a crow flies between two locations as opposed to the distance you travel in your car. The same idea holds in mathematics, which brings up the question: Is there always one way which produces a smallest or minimal distance in the mathematical world? In 1949 T. Motzkin answered this question and discovered a recursive method for determining values of a function which computes this distance, or more specifically, this “minimal” Euclidean norm; however, this recursive method becomes computationally intensive. Over the integers, a closed form for this norm has been found. Our work is centered on the closed form over the Eisenstein integers, or Z[?] where ? = (-1+?3 i)/2. This poster will show how we have analyzed the structure of residue classes modulo a+b?, how this has allowed us to reduce the number of necessary computations to find the minimal norm and describe how these results can be applied to determine the closed form over Z[?]. In addition, we will show a bit of code created to plot the values of the minimal norm.