An algebraic-perturbation variant of Barvinok's algorithm

We give a variant of Barvinok’s algorithm for computing a short rational generating function for the integer points in P := {x ∈ Rn : Ax ≤ b}; a use of which is to count the number of integer points in P . We use an algebraic perturbation, replacing each bi with bi + τ i, where τ > 0 is an arbitrarily small indeterminate. Hence, our new right-hand vector has components in the ordered ring Q[τ ] of polynomials in τ . Denoting the perturbed polyhedron by P (τ) ⊂ R[τ ]n, we use the facts that: P (τ) is full dimensional, simple, and contains the same integer points as P .