Extremal Completely Positive Plus-Rank

Definitions and mathematical statement

The supplied record does not define completely positive plus-rank or explicitly specify the field underlying $S^n$. Thus a fully self-contained statement requires clarification of these conventions.

Under the conventional real-symmetric interpretation, fix $n\in{\mathbb{Z}}_{\geq 1}$ and put \[ S^n_{{\mathbb{R}}}=\{A\in{\mathbb{R}}^{n\times n}:A^{\mathsf T}=A\}. \] A matrix $A\in S^n_{{\mathbb{R}}}$ is completely positive if $A=BB^{\mathsf T}$ for some entrywise nonnegative real matrix $B$. For nonzero $A$, its completely positive rank is \[ \operatorname{cp}(A)=\min\{r\in{\mathbb{Z}}_{\geq1}:\exists B\in{\mathbb{R}}_{\geq0}^{n\times r},\ A=BB^{\mathsf T}\}, \] with value $+\infty$ when no such factorization exists. This extended-value convention makes the finiteness restrictions explicit.

Write $q(A)=\operatorname{cp}^{+}(A)$ for the completely positive plus-rank intended by the source. Its defining admissible factorizations are absent from the supplied record and must be recovered from the cited references before $q$ can be given a definitive mathematical definition. In particular, no condition involving strictly positive factors is asserted here merely from the name.

Conditional on that definition and on the real-symmetric interpretation, define \[ \mathcal D_n=\{q(A)-\operatorname{cp}(A):A\in S^n_{{\mathbb{R}}}\setminus\{0\},\ q(A)<+\infty,\ \operatorname{cp}(A)<+\infty\} \] and \[ \mathcal P_n=\{q(A):A\in S^n_{{\mathbb{R}}}\setminus\{0\},\ q(A)<+\infty\}. \] The requested extremal quantities are \[ \max\mathcal D_n\qquad\text{and}\qquad\max\mathcal P_n, \] provided these maxima exist. The question asks for the greatest finite values occurring; it does not supply a proof of boundedness or attainment. The exclusion of zero follows the supplied cleaned statement.

There is a further quantifier ambiguity: the original wording does not explicitly fix the dimension for the first maximum. If that question ranges over all matrix sizes, its target is instead \[ \max\bigcup_{n\geq1}\mathcal D_n, \] again subject to existence. The fixed-dimension and unrestricted-dimension readings are retained as unresolved alternatives.

Short English statement

Determine the largest finite excess of completely positive plus-rank over completely positive rank, and the largest finite plus-rank in dimension $n$. The supplied record leaves the definition of plus-rank unspecified and does not settle whether the first maximum ranges over all dimensions; these issues require source review.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30003650$, title Extremal Completely Positive Plus-Rank, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the recorded proposed year is 2017.
• Supplied report citation: Copositivity and Complete Positivity (2018), Oberwolfach Reports, pp. 3071--3120, DOI https://doi.org/10.4171/owr/2017/52. The record attributes the problem to workshop open-problem discussions documented in this report.
• Supplied literature reference: Bomze--Dickinson--Still, "cp-rank and cp-plus-rank," https://doi.org/10.1016/j.laa.2015.05.021.
• Supplied literature reference: Oberwolfach Report 52, open questions, https://doi.org/10.4171/owr/2017/52.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the cited cp-plus-rank paper proves bounds and generic equality while leaving the maximal finite difference and exact maximal finite cp-plus-rank unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and the cited claims require independent verification.