Positive semigroups in lattices and totally real number fields
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Lenny Fukshansky
und
Siki Wang
Abstract
Let L be a full-rank lattice in ℝd and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+, and we investigate the distribution of the gaps of S(X) in L+, i.e. the points in L+ ∖ S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L+ and of L+ ∖ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.
Acknowledgements
We would like to sincerely thank the anonymous referee whose excellent suggestions helped to significantly improve our paper.
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Funding: Fukshansky was partially supported by the Simons Foundation grant #519058.
Communicated by: M. Henk
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Artikel in diesem Heft
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
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- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
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Artikel in diesem Heft
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
