Effective facet reconstruction from finite rational samples

Recover the primitive facet inequalities of an unknown rational polytope from finitely many values of the translated lattice-point enumerator L_{P+y*}(t) at positive rational dilation parameters, given bounds on the denominators, coordinates, and number of facets, and obtain an explicit upper bound for the required dilation parameters.

Background

The paper proves that, for the explicit vector y* = (2{1/(d+1)}, 2{2/(d+1)},..., 2{d/(d+1)}), the function L_{P+y*}(t) over positive rational t uniquely determines every full-dimensional rational polytope P. The proof is qualitative: each facet produces an infinite sequence of isolated discontinuities whose spacing is determined by its translated facet offset, and distinct facets yield incommensurable sequences.

The unresolved problem is to make this uniqueness result algorithmic and finite. Specifically, one must use bounded arithmetic and combinatorial data about P to recover its primitive facet inequalities from finitely many sampled counting-function values, while also explicitly bounding how large the dilation parameters need to be. The paper identifies effective control of facet relative inradii and covering radii of associated integral hyperplane lattices, together with a method for clustering observed jumps into facet sequences, as ingredients for such a result.

References

Given suitable bounds on the denominators, coordinates, and number of facets of an unknown rational polytope P, recover the primitive facet inequalities of P from finitely many values of L_{P+\mathbf y*}(t), \qquad t\inQ_{>0}, and obtain an explicit upper bound for the required dilation parameters.

— The integer point enumerator of one irrational translate of P is a complete invariant  (2608.19609 - Robins, 20 Aug 2026) in Section 6, Problem 1, “Effective facet reconstruction”

Suppose that, on a long finite interval, the discontinuities of two translated counting functions can be matched so that the corresponding jump times and jump sizes differ by prescribed small amounts. Under suitable a priori bounds on the rational polytopes, determine whether this forces quantitative closeness of their facet normals and facet offsets.

— The integer point enumerator of one irrational translate of P is a complete invariant  (2608.19609 - Robins, 20 Aug 2026) in Section 6, Problem 2, “Stability”