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

TL;DR

This paper introduces an algorithm to determine whether a prime ideal in a polynomial ring over complex numbers can be transformed into a toric ideal via a linear automorphism, and explicitly computes such transformations when possible.

Contribution

It presents a novel algorithm that effectively decides and constructs linear automorphisms transforming prime ideals into toric ideals, with applications to Gaussian graphical models and conditional independence ideals.

Findings

All Gaussian graphical models on five vertices not initially toric cannot be made toric by linear change.

All Gaussian conditional independence ideals on six vertices cannot be made toric by linear change.

The algorithm successfully computes explicit transformations when they exist.

Abstract

We propose an effective algorithm that decides if a prime ideal in a polynomial ring over the complex numbers can be transformed into a toric ideal by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. The algorithm can compute that all Gaussian graphical models on five vertices that are not initially toric cannot be made toric by any linear coordinate change. The same holds for all Gaussian conditional independence ideals of undirected graphs on six vertices.