##
**Lattice simplices of maximal dimension with a given degree.**
*(English)*
Zbl 1445.52010

Motivated by the problem of classifying lattice polytopes, that is, convex polytopes all of whose vertices lie in the integer lattice \(\mathbb{Z}^d\), with respect to natural parameters, the author studies the relationship of the dimension \(d\) of such a polytope \(P \subseteq \mathbb{R}^d\) and its degree \(s\).
The parameter \(s\) is just the degree of the \(h^*\)-polynomial \(h^*_P(t)\) of \(P\), defined by the well-known generating function identity
\[
\sum_{n \geq 0} |nP \cap \mathbb{Z}^d| \, t^n = \frac{h^*_P(t)}{(1-t)^{d+1}}.
\]
The starting point of the paper is a result of B. Nill [Eur. J. Comb. 29, No. 7, 1596–1602 (2008; Zbl 1149.52013)] who proved that the dimension of a lattice simplex that is not a lattice pyramid and whose degree is at most \(s\) is upper bounded by \(4s-2\).

The main result of the paper is a characterization of the extremal lattice simplices in this statement: A lattice simplex of degree \(s\) and dimension \(4s-2\) that is not a lattice pyramid is unimodularly equivalent to the simplex \(\Delta(r+2)\), where \(s=2^r\), \(r \in \mathbb{Z}_{\geq0}\), and where \(\Delta(r+2)\) is uniquely determined by \(r\) and the so-called \((r+2)\)-dimensional binary simplex code.

As a further insight the simplices \(\Delta(r+2)\) are shown to be counterexamples to what was known as the Cayley conjecture – a structural statement on the decomposition of a lattice polytope whose degree is smaller than half its dimension. A modified conjecture is offered for future studies.

The arguments in the proofs crucially rely on a characterization due to V. Batyrev and J. Hofscheier [“Lattice polytopes, finite abelian subgroups in \(\mathrm{SL}(n,\mathbb{C})\) and coding theory”, Preprint, arXiv:1309.5312] of lattice simplices that are not lattice pyramids in terms of an associated finite abelian group.

The main result of the paper is a characterization of the extremal lattice simplices in this statement: A lattice simplex of degree \(s\) and dimension \(4s-2\) that is not a lattice pyramid is unimodularly equivalent to the simplex \(\Delta(r+2)\), where \(s=2^r\), \(r \in \mathbb{Z}_{\geq0}\), and where \(\Delta(r+2)\) is uniquely determined by \(r\) and the so-called \((r+2)\)-dimensional binary simplex code.

As a further insight the simplices \(\Delta(r+2)\) are shown to be counterexamples to what was known as the Cayley conjecture – a structural statement on the decomposition of a lattice polytope whose degree is smaller than half its dimension. A modified conjecture is offered for future studies.

The arguments in the proofs crucially rely on a characterization due to V. Batyrev and J. Hofscheier [“Lattice polytopes, finite abelian subgroups in \(\mathrm{SL}(n,\mathbb{C})\) and coding theory”, Preprint, arXiv:1309.5312] of lattice simplices that are not lattice pyramids in terms of an associated finite abelian group.

Reviewer: Matthias Schymura (Lausanne)

### MSC:

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

94B05 | Linear codes (general theory) |

### Citations:

Zbl 1149.52013
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\textit{A. Higashitani}, Mich. Math. J. 68, No. 1, 193--210 (2019; Zbl 1445.52010)

### References:

[1] | V. V. Batyrev and J. Hofscheier, A generalization of a theorem of G. K. White, 2010, arXiv:1004.3411. |

[2] | V. V. Batyrev and J. Hofscheier, Lattice polytopes, finite abelian subgroups in \(\operatorname{SL}(n;\mathbb{C})\) and coding theory, 2013, arXiv:1309.5312. |

[3] | V. V. Batyrev and B. Nill, Multiples of lattice polytopes without interior lattice points, Mosc. Math. J. 7 (2007), 195-207. · Zbl 1134.52020 |

[4] | M. Beck and S. Robins, Computing the continuous discretely, Undergrad. Texts Math., Springer, 2007. · Zbl 1114.52013 |

[5] | S. Di Rocco, C. Haase, B. Nill, and A. Paffenholz, Polyhedral adjunction theory, Algebra Number Theory 7 (2013), no. 10, 2417-2446. · Zbl 1333.14010 |

[6] | A. Dickenstein and B. Nill, A simple combinatorial criterion for projective toric manifolds with dual defect, Math. Res. Lett. 17 (2010), no. 3, 435-448. · Zbl 1243.52010 |

[7] | C. Haase, B. Nill, and S. Payne, Cayley decompositions of lattice polytopes and upper bounds for \(h^*\)-polynomials, J. Reine Angew. Math. 637 (2009), 207-216. · Zbl 1185.52012 |

[8] | A. Higashitani and J. Hofscheier, Moduli spaces for lattice simplices with bounded degree, in preparation. |

[9] | A. Ito, Algebro-geometric characterization of Cayley polytopes, Adv. Math. 270 (2015), 598-608. · Zbl 1333.14048 |

[10] | B. Nill, Lattice polytopes having \(h^*\)-polynomials with given degree and linear coefficient, European J. Combin. 29 (2008), no. 7, 1596-1602. · Zbl 1149.52013 |

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