We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the satisfiability of the constraints over that field. This implies two complexity results about Gaussian conditional independence structures. First, all real algebraic numbers are necessary to construct inhabitants of non-empty Gaussian statistical models defined by conditional independence and dependence constraints. This gives a negative answer to a question of Petr \v{S}ime\v{c}ek. Second, we prove that the implication problem for Gaussian CI is polynomial-time equivalent to the existential theory of the reals.

中文翻译：

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通过对称矩阵的几乎主次态在射影平面上的入射几何

我们提出了一个将多项式系统编码成对称的，主要是规则的矩阵的几乎主次幂上消失和不消失的约束的方法，这样系统在某个字段上的可解性就等于该字段上约束的可满足性。这意味着关于高斯条件独立性结构的两个复杂性结果。首先，所有实数代数对于构造由条件独立性和依存关系约束定义的非空高斯统计模型的居民是必要的。这对Petr \ v {S} ime \ v {c} ek这个问题给出了否定的答案。其次，我们证明了高斯CI的蕴涵问题是多项式时间等效于实在论的存在论。