| 1. |
- Averkov, Gennadiy, et al.
(författare)
-
Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results
- 2015
-
Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 274, s. 118-166
-
Tidskriftsartikel (refereegranskat)abstract
- For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.
|
|
| 2. |
- Balletti, Gabriele
(författare)
-
Classifications and volume bounds of lattice polytopes
- 2017
-
Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume.In the first paper we give an upper bound on the volume vol(P^*) of a polytope P^* dual to a d-dimensional lattice polytope P with exactly one interiorlattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equivalence thereare 22,673,449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the h^*-polynomial in dimension three.
|
|
| 3. |
- Bogart, Tristram, et al.
(författare)
-
Finitely many smooth d-polytopes with n lattice points
- 2015
-
Ingår i: Israel Journal of Mathematics. - : Springer Science and Business Media LLC. - 0021-2172 .- 1565-8511. ; 207:1, s. 301-329
-
Tidskriftsartikel (refereegranskat)abstract
- We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙ n that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.
|
|
| 4. |
- Di Rocco, Sandra, et al.
(författare)
-
A note on discrete mixed volume and Hodge-Deligne numbers
- 2019
-
Ingår i: Advances in Applied Mathematics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0196-8858 .- 1090-2074. ; 104, s. 1-13
-
Tidskriftsartikel (refereegranskat)abstract
- Generalizing the famous Bernstein-Kushnirenko theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.
|
|
| 5. |
- Lorenz, Benjamin, et al.
(författare)
-
On smooth Gorenstein polytopes
- 2015
-
Ingår i: Tohoku mathematical journal. - : Mathematical Institute, Tohoku University. - 0040-8735. ; 67:4, s. 513-530
-
Tidskriftsartikel (refereegranskat)abstract
- A Gorenstein polytope of index r is a lattice polytope whose rth dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify d-dimensional smooth Gorenstein polytopes with index larger than (d + 3)/3. Moreover, we use a modification of Obro's algorithm to achieve classification results for smooth Gorenstein polytopes in low dimensions. The first application of these results is a database of all toric Fano d-folds whose anticanonical divisor is divisible by an integer r satisfying r >= d - 7. As a second application we verify that there are only finitely many families of Calabi-Yau complete intersections of fixed dimension that are associated to a smooth Gorenstein polytope via the Batyrev-Borisov construction.
|
|
