Vandermonde Cells Through the Lens of Positive Geometry

Abstract

We study the geometric and algebraic structure of Vandermonde cells, defined as images of the standard probability simplex under the Vandermonde map given by consecutive power sum polynomials. Motivated by their combinatorial equivalence to cyclic polytopes, which are well-known examples of positive geometries and tree amplituhedra, we investigate whether Vandermonde cells admit the structure of positive geometries. We derive explicit parametrizations and algebraic equations for their boundary components, extending known results from the planar case to arbitrary dimensions. By introducing a mild generalization of the notion of positive geometry, allowing singularities within boundary interiors, we show that planar Vandermonde cells naturally fit into this extended framework. Furthermore, we study Vandermonde cells in the setting of Brown-Dupont's mixed Hodge theory formulation of positive geometries, and show that they form a genus zero pair. These results provide a new algebraic and geometric understanding of Vandermonde cells, establishing them as promising examples within the emerging theory of positive geometries.