Counting Minimal Solutions to Diophantine Inequalities

Diophantine equations, those in which only integer solutions are of interest, have been studied since the 3rd century. Also of great interest are Diophantine inequalities. Indeed, these inequalities arose even in senior capstone projects. In 2013, Evan Cooper encountered a Diophantine inequality in his work to improve upon a college football ranking system. Cooper needed to determine the so-called minimal solutions to his inequality. Inspired by his work, we seek to determine the number of minimal solutions to a general linear Diophantine inequality. We begin by investigating the case of a two dimensional, linear, Diophantine inequality, and find a simple closed expression for the number of minimal solutions. For higher dimensional problems, we are able to determine a recursive formula for the number of minimal solutions. Finally, we make use of Ehrhart polynomials to find a closed form for both an upper and lower bound on the number of minimal solutions to a general linear Diophantine inequality in n variables.