Zero-Mutable Laurent Polynomials as Cluster Variables

Definitions and mathematical statement

For notation, if $A$ is a commutative coefficient ring, a Laurent polynomial in two variables is a finite sum \[ f(x,y)=\sum_{(a,b)\in S}c_{a,b}x^a y^b, \qquad S\subset{\mathbb{Z}}^2\text{ finite},\quad c_{a,b}\in A. \] The supplied record does not specify the intended coefficient ring; this notation does not choose one for the conjecture. A monomial has a single nonzero term, with any further restriction on its coefficient dependent on the intended convention.

The source describes a Laurent polynomial as zero-mutable if it permits a finite sequence of mutations ending in a monomial. Schematically, this means that there exist a finite integer $r\geq 0$ and permitted mutations $\mu_1,\ldots,\mu_r$ such that \[ f_0=f,\qquad f_j=\mu_j(f_{j-1})\quad(1\leq j\leq r), \qquad f_r\text{ is a monomial}. \] The permitted mutation operation, its admissibility conditions, and its coefficient conventions are not supplied. The displayed convention allows an empty sequence; the record does not explicitly discuss this boundary case.

In cluster algebra terminology, a cluster variable is a mutable variable occurring in a seed obtained from an initial seed by a finite sequence of seed mutations. Interpreting a Laurent polynomial as such a variable also requires specifying the cluster algebra's coefficients and an identification with the relevant Laurent expressions.

Conjecture as supplied, attributed to Gräfnitz--R. Every zero-mutable Laurent polynomial in two variables is a cluster variable for a suitable quiver. Its quantifier structure is \[ \text{for every zero-mutable }f\text{ in two variables, there exists a suitable quiver }Q \text{ for which }f\text{ is a cluster variable}. \] The quiver may depend on $f$. This is a formulation schema requiring clarification: the source record does not determine the coefficient domain, the Laurent-polynomial mutation operation, the allowed quivers and seed data, or the required identification of $f$ with a cluster variable. In particular, two Laurent variables do not, from the supplied wording alone, impose a rank-two restriction on $Q$. No normalization, positivity, support, or nondegeneracy hypothesis can be added from this record.

The original excerpt calls this a conjecture and adds "maybe soon a theorem." The supplied dated curation labels it open; neither qualification constitutes independent verification here.

Short English statement

The proposed assertion is that every two-variable Laurent polynomial that can be mutated into a monomial occurs as a cluster variable for some suitable quiver. The coefficient conventions, allowed mutations, and meaning of the required cluster realization remain unspecified, so this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006444$, title Zero-Mutable Laurent Polynomials as Cluster Variables, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2025$, leaves the proposer field empty, and attributes the conjecture within the original excerpt to "Gräfnitz--R." The abbreviated attribution is retained without expansion.
• Supplied source citation: Singularities (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/45, pp. 2413--2490. https://doi.org/10.4171/owr/2025/45. The citation's title year differs from the report year in the DOI and other supplied metadata; this discrepancy is unresolved.
• Supplied literature source: Oberwolfach Report 45/2025. https://oa.tib.eu/renate/bitstreams/e0847358-34cd-4809-8330-e99d50f1e37e/download.
• Supplied literature source: Smoothings from zero-mutable Laurent polynomials. https://arxiv.org/abs/2503.18661.
• Literature assessment dated 2026-08-22: the supplied curation labels the universal implication open and still conjectural, and describes related work as developing zero-mutable polynomials and proving important families of associated smoothing results, rather than the universal cluster-variable assertion. This is supplied dated curation, not an independently verified literature assessment. No linked source was visited for this draft.