On the Volume and Integral Points of a Polyhedron in Rn: Basics, Concepts, Methods

Computing the volume and integral points of a polyhedron in Rn is a very important subject in different areas of mathematics. Chapter one give a short introduction, provide a sketch of what polytope looks like and how they behave with many examples. Some methods for finding the number of integral points inside a convex polytope are recalled. Chapter two, present some methods for computing the coefficients of Ehrhart polynomial that depend on the concepts of Dedekind sum and residue theorem in complex analysis. We give a method for computing the coefficients of the Ehrhart polynomial, cd-3, cd-4 until cd-9 also a formula for the differentiation of the given method is introduced. Chapter three, a method for finding the volume of a polytope using La?lace transform is presented and some basic concepts and remarks about the Birkhoff polytope and their volumes are discussed with their Ehrhart polynomials. We find a general formula for the number of integral points. To the best of our knowledge, this result seems to be new. The analysis should be useful to professionals in combinatorial, algebraic geometry, cryptography, and integer programming.