Sail formulation of Oppenheim's conjecture

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Let n⩾3n\geqslant 3, let Λ⊂Rn\Lambda\subset\mathbb{R}^n be an nn-dimensional lattice, and let Π\Pi be a sail generated by Λ\Lambda. A sail's facets and edge stars are its facets and the edge stars of its vertices; they have uniformly bounded determinants when these determinants are bounded by a constant independent of the facet or edge star. Sail formulation of Oppenheim's conjecture. If all facets and edge stars of vertices of Π\Pi have uniformly bounded determinants, then Λ\Lambda is algebraic.

The paper derives this formulation from its theorem equating positive norm minimum with uniform determinant bounds. It is therefore equivalent to Oppenheim's conjecture and remains open.

References

Primary source

Oleg N. German, “Klein polyhedra and lattices with positive norm minima”, arXiv:math/0504483 (2006).

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