Local Ehrhart Theory and its Synergies

• Applicant: Professor Dr. Benjamin Nill
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539867500

The study of lattice polytopes (polytopes whose vertices have integer coordinates), is a very active area of algebraic, enumerative and geometric combinatorics where many different disciplines meet in a fruitful and synergetic way: discrete, convex and metric geometry, geometry of numbers, linear optimization, algebraic geometry, commutative algebra, symplectic geometry, theoretical physics, … : Ehrhart theory of lattice polytopes can be described as the study of the h*-polynomial, a linear transformation of the Ehrhart polynomial: the fundamental invariant that counts lattice points in dilates of lattice polytopes. Understanding the coefficients of h*-polynomials of lattice polytopes is one of the main goals in this area of research with numerous results over the last decades. As Karu and Stanley showed, the h*-polynomial can be decomposed into nonnegative local contributions of faces using toric g-polynomials. The main local contribution from the polytope itself is called local h*-polynomial in a fundamental article by Katz and Stapledon. It had already been in use under a different name by Borisov, Batyrev, Mavlyutov, Schepers in the computation of stringy Hodge numbers of Calabi-Yau complete intersections and has also been known as the box polynomial in the case of lattice simplices. In a recent paper using methods from the groundbreaking proof of the g-theorem Adiprasito, Papadakis, Petrotou, and Steinmeyer show that the local h*-polynomial of IDP polytopes has a unimodal coefficient vector. With this project we establish local Ehrhart theory as the fertile field of study of local h*-polynomials of lattice polytopes and its rich and diverse synergies. We plan to investigate the complexity of the space of local h*-vectors and to explore polytopal constructions that leave local h*-polynomials invariant. One particularly fascinating phenomenon is that local h*-polynomials can be zero. Such lattice polytopes are called thin polytopes, and were first investigated by Gelfand, Kapranov and Zelevinsky. We plan to work on finding combinatorial and geometric restrictions on thin polytopes and to investigate an interesting natural class of thin polytopes. It is our strong intention to exploit and enhance existing and new relations of local Ehrhart theory with hypergeometric motives, mirror symmetry, dual defective varieties, coding theory, hyperplane arrangements, metric geometry, neighborly polytopes, and TU-matrices among others. Our endeavor is heavily data-driven. We plan to generate databases of lattice polytopes and their local h*-polynomials and to classify lattice polytopes with special local-Ehrhart-theoretic properties. One major aspect is the application of machine-learning algorithms to explore whether invariants and properties in (local) Ehrhart theory can be efficiently computed and detected. Our underlying goal is to use these tools as guides in computer experiments looking for interesting examples and counterexamples.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies