Bounds of Linear Functions

Abstract

We saw in Chapter 3 that to determine if two variables caused a dependence between two statements, with a given direction vector (or at a given level), one must decide if there is an integer solution to a system of linear diophantine equations satisfying a system of linear inequalities (constraints). If the system of equations (without any further constraints) has no integer solution, then there is no dependence. When integer solutions (to the system of equations) are known to exist, there are basically two approaches at that point: find the general solution and see if it can be tailored to fit the constraints, or check certain necessary conditions that must hold if a solution satisfying all constraints is to exist. In Chapter 5, we would lay the groundwork for the first approach by showing how to solve linear diophantine equations. This chapter prepares us for the second approach. Here, the actual expression for the general solution is not needed; we use necessary conditions involving bounds of the linear functions that represent the left hand sides of the equations.