A New Algorithm in Geometry of Numbers
Mathieu Dutour, Konstantin Rybnikov
TL;DR
This paper introduces a novel algorithm for identifying perfect Delaunay polytopes in lattice geometry, overcoming previous limitations and enabling discoveries in higher dimensions, which may influence the understanding of lattice Delaunay tilings.
Contribution
The paper presents a new algorithm that improves the process of finding perfect Delaunay polytopes, facilitating exploration in dimensions 6 to 8.
Findings
Successful implementation in dimensions 6, 7, 8
Discovery of a new conjecture about lattice Delaunay tilings
Enhanced algorithm performance over previous methods
Abstract
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.
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