射影平面上对称矩阵的几乎-主子式的关联几何
Incidence geometry in the projective plane via almost-principal minors of symmetric matrices
摘要：
我们提出了一个编码的多项式系统为零和非零的几乎-主子式的对称，主要是正则矩阵的约束，使系统的可解性在某些领域是等价的满足该领域的约束。这意味着高斯条件独立结构的两个复杂性结果。首先，所有的实代数数是必要的，以构建居民的非空高斯统计模型定义的条件独立性和依赖性约束。这对彼得·希梅切克的一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价于实数的存在性理论。
英文摘要：
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 Šimeček. Second, we prove that the implication problem for Gaussian CI is polynomial-time equivalent to the existential theory of the reals.
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