A New Algorithm in Geometry of Numbers

A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings.

💡 Research Summary

The paper addresses the long‑standing problem of efficiently identifying perfect Delaunay polytopes—lattice polytopes whose Delaunay sphere is the unique ellipsoid circumscribing them. Such polytopes are fundamental building blocks in the theory of lattice Delaunay tilings and have deep connections to integer optimization, sphere packing, and the geometry of numbers. Existing approaches, largely based on global linear programming and exhaustive enumeration of lattice candidates, become computationally infeasible already in moderate dimensions because the number of potential lattice configurations grows exponentially with the dimension. Moreover, these methods often fail to exploit the rich symmetry inherent in lattice structures, leading to redundant checks and an inability to certify perfection for dimensions five and higher.

The authors propose a two‑stage algorithm that dramatically reduces the search space and accelerates the verification of perfection. In the first stage, they fix the equation of the Delaunay sphere (or, more generally, the circumscribing ellipsoid) and use group‑theoretic reduction techniques. By computing the automorphism group of the underlying lattice and converting each candidate lattice to a canonical normal form, they partition the set of possible lattices into equivalence classes under the action of the symmetry group. This eliminates duplicate investigations of congruent configurations and shrinks the candidate pool from exponential to polynomial growth in the dimension.

The second stage replaces the costly global linear program traditionally used to test whether a given polytope is perfect. Instead, the authors introduce a local quadratic‑form verification procedure. For each facet of a candidate polytope they compute the outward normal vector and check the tangency condition with the fixed Delaunay sphere simultaneously for all facets. This reduces the verification to solving a small system of quadratic equations whose size is proportional to the number of facets, yielding a polynomial‑time test. The authors provide a rigorous complexity analysis showing that, for dimension d, the overall algorithm runs in O(poly(d)) time for candidate generation and O(f·d) time for perfection testing, where f is the number of facets.

Implementation details are described in depth. The authors built a software pipeline that integrates SageMath for lattice and group computations with PARI/GP for high‑precision quadratic‑form arithmetic. They parallelized the search across multiple cores, assigning each equivalence class to a separate worker. The code automatically filters out non‑full‑dimensional candidates, computes the canonical normal form, and then applies the quadratic‑form test.

Experimental results are reported for dimensions six, seven, and eight. In dimension six the algorithm rediscovered all three previously known perfect Delaunay polytopes, confirming correctness. In dimension seven it uncovered two new perfect polytopes that exhibit a non‑trivial symmetry group isomorphic to a direct product of the symmetric group S₇ and a dihedral component. In dimension eight a single new perfect polytope was found; its automorphism group displays a hierarchical extension of the symmetry observed in lower dimensions, a pattern not captured by earlier methods. All newly discovered polytopes satisfy the strict definition of perfection, and their existence demonstrates that the search space previously considered exhaustive was in fact incomplete.

Analyzing the symmetry structures of the newly found polytopes, the authors formulate a conjecture: the automorphism groups of perfect Delaunay polytopes in successive dimensions form a nested hierarchy, where the group in dimension d contains the group in dimension d‑1 as a normal subgroup extended by a well‑defined set of additional symmetries. This conjecture, if true, would impose strong constraints on the possible configurations of lattice Delaunay tilings in high dimensions and could guide future searches for perfect polytopes beyond dimension eight.

The paper concludes by emphasizing the algorithm’s scalability, its potential to be adapted for dimensions nine and higher, and the broader implications for the geometry of numbers. The authors suggest several avenues for future work: formal proof or refutation of the hierarchical symmetry conjecture, extension of the method to other classes of lattice polytopes (e.g., empty lattice simplices), and application of the discovered perfect polytopes to improve bounds in sphere‑packing and covering problems. Overall, the work represents a significant methodological advance and opens new directions in the study of lattice Delaunay structures.