Recursive Degree Formulas in Likelihood Geometry

Definitions and mathematical statement

For each integer $n\geq 1$, the supplied record refers to the unrestricted $n\times n$ correlation model. The conventional real parameter space for such a model is \[ \mathcal C_n=\{\Sigma\in{\mathbb{R}}^{n\times n}:\Sigma^{\mathsf T}=\Sigma,\ \Sigma\text{ is positive definite},\ \Sigma_{ii}=1\text{ for }1\leq i\leq n\}. \] This conventional interpretation is stated explicitly as a qualification: the supplied record does not define the model's algebraic realization, its treatment of singular boundary points, or the degree invariants below. In particular, this parameter-space description does not supply the missing definitions of those invariants.

Write \[ a_n=\operatorname{SSLdeg}(\mathcal C_n),\qquad b_n=\operatorname{MLdeg}(\mathcal C_n),\qquad c_n=\operatorname{dualMLdeg}(\mathcal C_n) \] as placeholders for the source's intended SSL degree, ML degree, and dual ML degree, respectively. The supplied record does not expand SSL or specify the objective functions, critical-point equations, generic data, complex counting domains, excluded loci, or multiplicity conventions defining these numbers. Consequently, these symbols name the intended invariants but do not independently define them.

Problem. With the source's intended definitions of these three degrees, find recursive formulas determining $(a_n)_{n\geq1}$, $(b_n)_{n\geq1}$, and $(c_n)_{n\geq1}$ for the unrestricted correlation models. The original excerpt asks for a formula, with recursion suggested parenthetically; the cleaned registry statement specifically requests recursive formulas. The supplied initial values are \[ \begin{array}{c|rrrrrrrrr} n&1&2&3&4&5&6&7&8&9\\\hline a_n&1&4&28&292&?&?&?&?&?\\ b_n&1&3&15&109&1077&13695&?&?&?\\ c_n&1&2&5&14&43&144&522&2028&8357 \end{array} \] Here a question mark denotes an entry left unspecified in the supplied table. These finite lists alone do not define the sequences. A fully specified mathematical formulation requires the source definitions and counting conventions; no recurrence or continuation is asserted here.

Short English statement

Find recursive formulas for the SSL degree, ML degree, and dual ML degree of unrestricted correlation models of every size $n\geq1$, consistent with the supplied table. The record omits the definitions of these degrees, so the intended counting problems require clarification before this can be a fully self-contained statement. The supplied open status is a dated curation label, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30005309$, titled Recursive Degree Formulas in Likelihood Geometry; exact source code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and assigns proposed year 2022. No individual proposer is identified.
• Supplied report citation: Algebraic Structures in Statistical Methodology (2023), Oberwolfach Reports, pp. 3121--3170, DOI 10.4171/owr/2022/55. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2022/55. The record describes this report as documenting workshop discussions and open problems.
• Supplied literature-source link labeled "Amendola--Zwiernik primary article": https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/2274/1147/6553. The supplied record gives no article title or further bibliographic details.
• Supplied literature assessment, dated exactly 2026-08-22: the status is labeled open; the degree table is described as incomplete, recursive formulas as still open, and the cited likelihood-geometry work as supplying only small cases. This is supplied dated curation, described in the record as literature triage derived from the OWR curation, and has not been independently verified here. Time-sensitive status and the cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired. These are supplied provenance claims, not verification performed for this draft.