Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures of Sturmfels and Uhler from \cite{BS n CU}. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree $1$ and $2$. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the $1$-clique sums of complete graphs.

中文翻译：

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具有复曲面消失理想的高斯图模型

高斯图模型是正定协方差矩阵锥的半代数子集。它们广泛应用于自然科学、计算生物学和许多其他领域。计算模型的消失理想为我们提供了模型的隐式描述。在本文中，我们从 \cite{BS n CU} 中解决了 Sturmfels 和 Uhler 的两个猜想。特别地，我们描述了那些以 $1$ 和 $2$ 的度数生成高斯图形模型的消失理想的图。这些结果是高斯图模型，其理想是复曲面理想，结果图是完整图的 $1$-clique 和。