Combining Realization Space Models of Polytopes

In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of Criado and Santos (Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1641766), about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them.

Automated summary

This paper examines four different models for the realization space of a polytope and explores their relationships and how to combine the strengths of different models. By combining the compact nature of the Grassmannian model with the slack variety, a reduced slack model is obtained, which further expands the research on the realization of polytopes.