Counting the number of integer points of non-parametric and parametric polytopes with the PolyhedralSets in Maple

When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints this task may simply become an undecidable problem. However, studying the integer solutions of systems of equations and inequalities is of practical importance in areas like combinatorial optimization and compiler construction. Over the past 10 years, a series of projects has equipped the computer algebra system Maple with a number of tools for dealing with the integer points of systems of linear equations and inequalities, even in the presence of parameters. With these tools, one can either decide whether integer point solutions exist, or count them, or describe them in a compact way, or enumerate them. In this paper, we will give a tour of these facilities and illustrate their usage with a number of applications. In this paper, we have discussed the adaptation of Barvinok’s algorithm for integer point counting from non-parametric polytopes to parametric polytopes. We report on a Maple implementation of this adaptation. This includes a general framework called Value- UnderConstraints in support of parametric system solving. Additionally, in order to represent the parametric results of these counting of para- metric system, a new Maple data structure called Quasi-polynomial was built.