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+τi, 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.