On lattice point counting in $$\varDelta $$-modular polyhedra

Let a polyhedron P be defined by one of the following ways: and let all rank order minors A bounded $$\varDelta $$ in absolute values. We show that short rational generating function for power series $$\begin{aligned} \sum \limits _{m \in \cap {{\,\mathrm{{\mathbb {Z}}}\,}}^n} {{\,\mathrm{{\mathbf {x}}}\,}}^m \end{aligned}$$ can computed with arithmetical complexity O\left( T_{{\mathrm{SNF}}}(d) \cdot d^{k} d^{\log _2 \varDelta }\right) , where k are fixed, $$d = \dim P$$ $$T_{{\mathrm{SNF}}}(m)$$ is computing Smith Normal Form $$m \times m$$ integer matrices. In particular, n$$ case (i), n-k$$ (ii). The simplest examples polyhedra meet conditions (i) or (ii) simplices, subset sum polytope knapsack multidimensional polytopes. Previously, existence polynomial time algorithm varying dimension considered class problems was unknown already simplicies ( $$k 1$$ ). apply these results to parametric polytopes step representation $$c_P({{\,\mathrm{{\mathbf {y}}}\,}}) |P_{{{\,\mathrm{{\mathbf {y}}}\,}}} {Z}}}\,}}^n|$$ $$P_{{{\,\mathrm{{\mathbf {y}}}\,}}}$$ polytope, whose structure close cases (ii), even if not fixed. As another consequence, we coefficients $$e_i(P,m)$$ Ehrhart quasi-polynomial \left| mP {Z}}}\,}}^n\right| _{j 0}^n e_j(P,m)m^j polynomial-time algorithm, fixed .