Efficiently deciding if an ideal is toric after a linear coordinate change

We propose an effective algorithm to decide if a homogeneous prime ideal can be made toric by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. We benchmark the algorithm on Gaussian graphical models on five vertices and Gaussian conditional independence models of undirected graphs up to six vertices. We find that these are either toric from the start or cannot be made toric by linear coordinate changes. This is joint work with Julian Vill.