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.


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)


Zbl 1149.52013
Full Text: DOI arXiv


[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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