Combinatorial Theory

The local \(h^*\)-polynomial is a natural invariant of a lattice polytope appearing in Ehrhart theory and Hodge theory. In this work, we study the question posed by Gelfand-Kapranov-Zelevinsky in 1994 concerning the classification of lattice simplices with vanishing local \(h^*\)-polynomial. Such simplices are called thin. We relate this question to linear codes and hyperplane arrangements over finite rings. This allows us to obtain a complete classification of the \(4\)-dimensional thin simplices, extending the previously known results in dimensions up to \(3\).

Mathematics Subject Classifications: 52B20, 94B05, 52C35

Keywords: Lattice simplex, Ehrhart theory, local \(h^*\)-polynomial, linear code, hyperplane arrangement