Alternating Cyclic-Quotient Identities over Commutative Fields

Definitions and mathematical statement

Let $K$ be a commutative field. The supplied record uses a bracket $[AB]$ for $A,B\in K^2$, but does not define it. Consequently, the following is a conditional transcription of the extracted question, requiring a specified $K$-valued bracket before it becomes a fully determined mathematical statement.

For a positive integer $n$, let $A_1,\ldots,A_n,A'_1,\ldots,A'_n\in K^2$. Write $S_n$ for the permutations of $\{1,\ldots,n\}$, and set \[ C_n=\{\pi\in S_n:\pi(1)=1\}. \] For $\pi\in C_n$, put $\pi(n+1)=\pi(1)$. Provided that \[ D_\pi=\prod_{i=1}^n[A'_{\pi(i)}A'_{\pi(i+1)}]\ne0 \qquad\text{for every }\pi\in C_n, \] define \[ \alpha_\pi= \alpha(A_{\pi(1)},\ldots,A_{\pi(n)}\mid A'_{\pi(1)},\ldots,A'_{\pi(n)}) =\frac{\displaystyle\prod_{i=1}^n[A_{\pi(i)}A_{\pi(i+1)}]}{\displaystyle\prod_{i=1}^n[A'_{\pi(i)}A'_{\pi(i+1)}]}. \] The extracted question asks whether, for every commutative field $K$, every intended value of $n$, and every configuration for which all these quotients are defined, \[ \sum_{\pi\in C_n}\operatorname{sgn}(\pi)\,\alpha_\pi=0 \quad\text{in }K, \] where the integer signs are interpreted through the canonical map ${\mathbb{Z}}\to K$.

The denominator condition above is necessary to interpret the displayed quotients in a field; it is not stated in the supplied extraction. The intended bracket and range of $n$, including the treatment of small values and self-brackets, remain unspecified. No characteristic restriction is supplied. The supplied background explicitly describes the assertion as a truncated cyclic-quotient conjecture, so additional hypotheses or qualifications may be missing.

Short English statement

The extracted question asks whether the signed sum of cyclic bracket-product quotients vanishes when the same permutation reorders both lists and fixes the first index. A precise conjecture requires the missing bracket definition, the intended range of $n$, and confirmation of the admissibility conditions. The supplied record labels the problem open, but this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json and supplementary catalog/problems.json; supplied record for permanent UnsolvedMath ID $30001137$, entitled Alternating Cyclic-Quotient Identities over Commutative Fields, exact source problem code catalogue entry. These registry references identify the supplied record; their file contents have not been independently inspected here.
• Supplied report citation: Discrete Differential Geometry (2009), Oberwolfach Reports, pp. 75--144, DOI 10.4171/owr/2009/02. Original report link: https://doi.org/10.4171/owr/2009/02. The record attributes the collection to workshop open-problem discussions and supplies no individual proposer.
• Supplied literature reference: Anderson--Delucchi, complex matroids, https://arxiv.org/abs/1005.3560.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the extracted assertion as a truncated cyclic-quotient conjecture whose general formula is not established by the checked complex-matroid literature. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.