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

Higashitani, Akihiro

Mich. Math. J. 68, No. 1, 193-210 (2019).

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.

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)

Keywords:

lattice simplices; degree; Cayley conjecture; binary simplex codes

Citations:

Zbl 1149.52013

References:

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