Thin Simplices via Modular Arithmetic

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 in GKZ94 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.