Preserving Hodge Vectors of Lattice Polytopes

In this talk we will focus on the local $h^*$-polynomial of a lattice polytope and its coefficient list, the Hodge vector. This vector arises from the top-degree part of a certain Hodge-Deligne polynomial of an associated generic hypersurface in an algebraic torus. We introduce a construction that preserves the Hodge vector while generating higher-dimensional polytopes, providing an Ehrhart-theoretic analogue of the classical reduction formula for the mixed volume. Additionally, this technique yields a formula for the $h^*$-polynomial of the projection of a lattice polytope along a ray. This is joint work with Benjamin Nill.