Polynomial-Time Lattice-Point Counting without Barvinok Decomposition

Abstract

By using constant term manipulations, we present the first polynomial-time algorithm for lattice-point counting in fixed dimension that does not rely on Barvinok's unimodular decomposition. The algorithm instead operates directly on a rational generating function in the form of a nested root average, as produced by the \texttt{SimpCone[S]} framework. By means of a residue-lattice argument based on Minkowski's theorem, we construct a short multiplier that induces an exact non-coprime split of the outermost average. The resulting child terms are encoded as joint root averages, and Smith normal form is used to restore the recursive structure. Two structural invariants---the generation condition and full-column independence---ensure that the recursion is well defined and that all required pole exchanges are valid. For a fixed-dimensional simplicial cone, the algorithm achieves recursion depth \(O_d(1+\log\log(2+\ind(\mathcal K^*)))\) and produces a signed sum of at most \((1+\log \ind(\mathcal K^*))^{O_d(1)}\) unimodular cone generating functions. The framework uniformly handles numerators that are Laurent polynomials, not merely monomials, thereby giving a polynomial-time algorithm for MacMahon's partition analysis when the dimension is fixed.