Lattice Size of Width One Lattice Polytopes in $\mathbb{R}^3$
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| Format: | Preprint |
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2022
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| _version_ | 1866916252852682752 |
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| author | Alajmi, Abdulrahman Soprunova, Jenya |
| author_facet | Alajmi, Abdulrahman Soprunova, Jenya |
| contents | The lattice size $\operatorname{ls_Δ}(P)$ of a lattice polytope $P$ is a geometric invariant, which was formally introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics. In this paper, we show that for an empty lattice polytope $P\subset\mathbb{R}^3$ there exists a reduced basis of $\mathbb{Z}^3$ which computes its lattice size $\operatorname{ls_Δ}(P)$. This leads to a fast algorithm for computing $\operatorname{ls_Δ}(P)$ for such $P$. We also extend this result to another class of lattice width one polytopes $P\subset\mathbb{R}^3$. We then provide a counterexample demonstrating that this result does not hold true for an arbitrary lattice polytope $P\subset\mathbb{R}^3$ of lattice width one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_13124 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lattice Size of Width One Lattice Polytopes in $\mathbb{R}^3$ Alajmi, Abdulrahman Soprunova, Jenya Combinatorics 11H06, 52B20, 52C05 The lattice size $\operatorname{ls_Δ}(P)$ of a lattice polytope $P$ is a geometric invariant, which was formally introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics. In this paper, we show that for an empty lattice polytope $P\subset\mathbb{R}^3$ there exists a reduced basis of $\mathbb{Z}^3$ which computes its lattice size $\operatorname{ls_Δ}(P)$. This leads to a fast algorithm for computing $\operatorname{ls_Δ}(P)$ for such $P$. We also extend this result to another class of lattice width one polytopes $P\subset\mathbb{R}^3$. We then provide a counterexample demonstrating that this result does not hold true for an arbitrary lattice polytope $P\subset\mathbb{R}^3$ of lattice width one. |
| title | Lattice Size of Width One Lattice Polytopes in $\mathbb{R}^3$ |
| topic | Combinatorics 11H06, 52B20, 52C05 |
| url | https://arxiv.org/abs/2207.13124 |