[Paper Review] Gorenstein polytopes with trinomial $h^*$-polynomials

Akihiro Higashitani, Benjamin Nill+1 more|arXiv (Cornell University)|Mar 19, 2015

Advanced Combinatorial Mathematics25 references3 citations

TL;DR

This paper classifies Gorenstein lattice polytopes whose $h^*$-polynomials are trinomials (i.e., have exactly three non-zero terms), completing a classification initiated by Batyrev and Juny for degree-two Gorenstein polytopes. By leveraging the characterization of empty simplices with binomial $h^*$-polynomials by Batyrev and Hofscheier, the authors fully determine all such trinomial $h^*$-polynomials and their corresponding polytopes, providing a complete classification in all dimensions.

ABSTRACT

The characterization of lattice polytopes based upon information about their Ehrhart $h^*$-polynomials is a difficult open problem. In this paper, we finish the classification of lattice polytopes whose $h^*$-polynomials satisfy two properties: they are palindromic (so the polytope is Gorenstein) and they consist of precisely three terms. This extends the classification of Gorenstein polytopes of degree two due to Batyrev and Juny. The proof relies on the recent characterization of Batyrev and Hofscheier of empty lattice simplices whose $h^*$-polynomials have precisely two terms. Putting our theorem in perspective, we give a summary of these and other existing results in this area.

Motivation & Objective

• To complete the classification of Gorenstein lattice polytopes whose $h^*$-polynomials have exactly three non-zero terms.
• To extend previous results on Gorenstein polytopes of degree two and empty simplices with binomial $h^*$-polynomials.
• To characterize all lattice polytopes whose $h^*$-polynomials are palindromic trinomials, leveraging structural constraints from Ehrhart theory and lattice point enumeration.
• To provide a comprehensive summary of existing results on $h^*$-polynomials with few terms, placing the current work in context.

Proposed method

• Utilize the characterization of empty lattice simplices with binomial $h^*$-polynomials by Batyrev and Hofscheier as a foundational input.
• Apply the theory of lattice pyramids, noting that $h^*$-polynomials are invariant under pyramid construction.
• Analyze the structure of the $h^*$-polynomial via the normalized volume and degree, focusing on palindromic trinomials.
• Use group-theoretic and combinatorial techniques on the semigroup of lattice points in the polytope to classify possible configurations.
• Employ case analysis based on the order of generators in the class group $\Lambda_\Delta$ to enumerate all possible polytope types.
• Verify that all cases lead to exactly three non-zero coefficients in the $h^*$-polynomial, confirming the trinomial condition.

Experimental results

Research questions

• RQ1 Which Gorenstein lattice polytopes have $h^*$-polynomials with exactly three non-zero terms?
• RQ2 How can the classification of empty simplices with binomial $h^*$-polynomials be extended to Gorenstein polytopes with trinomial $h^*$-polynomials?
• RQ3 What are the complete list of possible trinomial $h^*$-polynomials that arise from lattice polytopes, and what are the corresponding polytope types?
• RQ4 Can all such trinomial $h^*$-polynomials be realized as the $h^*$-polynomial of a Gorenstein polytope, and if so, what are their geometric and algebraic properties?

Key findings

• All Gorenstein lattice polytopes with trinomial $h^*$-polynomials are completely classified, completing the program initiated by Batyrev and Juny.
• The $h^*$-polynomials are of the form $1 + at + bt^d$ with $a,b \in \mathbb{Z}_{\geq 0}$, and the classification depends on the structure of the class group $\Lambda_\Delta$.
• The classification includes lattice pyramids over specific polygons and Cayley polytopes, with explicit constructions provided.
• The authors identify three distinct families of polytopes corresponding to the cases where the class group $\Lambda_\Delta$ is isomorphic to $({\mathbb{Z}}/3{\mathbb{Z}})^2$, $({\mathbb{Z}}/2{\mathbb{Z}})^2$, or $({\mathbb{Z}}/4{\mathbb{Z}})^2$, depending on the order of generators.
• The case where the class group has elements of order 6 leads to a contradiction, showing that such configurations cannot yield trinomial $h^*$-polynomials.
• The results confirm that the only possible trinomial $h^*$-polynomials for Gorenstein polytopes are those with $h^*_0 = h^*_d = 1$, $h^*_1 = a$, and $h^*_d = b$, with constraints derived from the geometry of the polytope and its lattice point structure.