Arithmetic sparsity in mixed Hodge settings

Abstract Let be a smooth irreducible quasi‐projective algebraic variety over a number field . Suppose is equipped with a ‐adic étale local system compatible with an admissible graded‐polarized variation of mixed Hodge structures on the complex analytification of . We prove that the ‐integral points in are covered by subpolynomially many geometrically irreducible ‐subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe–Maculan and Ellenberg–Lawrence–Venkatesh. As an application, we prove that there are subpolynomially many ‐integral Laurent polynomials with fixed reflexive Newton polyhedron and fixed non‐zero principal ‐determinant. Our results answer a question asked by Ellenberg–Lawrence–Venkatesh.