Open Problems in Algebraic Statistics — Bonus Problem 8

Definitions and mathematical statement

The supplied source asks whether every strict CI model has a ${\mathbb{Q}}$-rational point, where CI denotes conditional independence. The record does not define the class of models or the qualification strict.

For a model $M$ with specified ambient coordinates, write \[ M({\mathbb{Q}})=\{p\in M: \text{every ambient coordinate of }p\text{ belongs to }{\mathbb{Q}}\}. \] Here membership in $M$ must include all conditions defining the intended strict model. Let $\mathscr S$ denote the class intended by the source. The question is the following statement schema: \[ \forall M\in\mathscr S,\qquad M({\mathbb{Q}})\ne\varnothing. \] This is not a fully specified mathematical assertion until $\mathscr S$, the ambient coordinates, and the meaning of strictness are supplied. In particular, the record does not specify the random-variable domains, the probability-model framework, whether models are required to be nonempty, or which equalities and inequalities define their admissible points. No positivity condition or interpretation of strictness as the absence of additional conditional independences is assumed in this draft.

Short English statement

Does every strict conditional-independence model contain an admissible point with rational coordinates? The supplied record leaves both the intended class of models and the meaning of strict undefined, so the displayed formulation remains a statement schema.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $800869$, exact source code catalogue entry, titled Open Problems in Algebraic Statistics --- Bonus Problem 8. The registry attributes the problem to Sturmfels, gives the proposed year as 2007, and labels it open; this label is supplied metadata, not an independently verified mathematical or literature assessment.
• Supplied source attribution: Sturmfels, Open Problems in Algebraic Statistics (2007), Bonus Problem (#8), https://arxiv.org/abs/0707.4558. The background records access on 2026-10-16 and extraction by arxiv_latex. These provenance details are supplied by the record and have not been independently verified here. The complete referenced work was not supplied.