Sullivant-Talaska ideal of the cyclic Gaussian Graphical Model

TL;DR

This paper proves a conjecture about the algebraic structure of Gaussian graphical models for cycle graphs, providing a general method applicable to similar ideals with radical initial ideals.

Contribution

It settles a longstanding conjecture on the prime ideal generation for cyclic Gaussian graphical models and introduces broadly applicable methods.

Findings

Confirmed the conjecture for all cycle graphs

Developed a general approach for ideals with radical initial ideals

Enhanced understanding of algebraic properties of Gaussian graphical models

Abstract

In this paper, we settle a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph. Our methods are general and applicable to a large class of ideals with radical initial ideals.