Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of geometric triangulations of simplices whose local h-polynomial vanishes. As a first step, we identify a class of refinements that preserve the local h-polynomial. In dimensions 2 and 3, we show that all geometric triangulations with vanishing local h-polynomial are obtained from one or two simple examples by a sequence of such refinements. In higher dimensions, we prove some partial results and give further examples.i are nonnegative and satisfy i = d−i . Moreover, if the subdivision is regular, then these coefficients are unimodal. Among other applications, Stanley used local h-polynomials to prove that h-polynomials increase coefficientwise under refinement.As discussed in Section 2, the local h-polynomial also behaves predictably with respect to basic operations on subdivisions. It is additive for refinements that nontrivially subdivide only one facet, multiplicative for joins, and vanishes on the trivial subdivision. In particular, if Γ is a refinement of Γ that nontrivially subdivides only Manuscript