Realizability of Balanced Complexes as Tropical Varieties

Definitions and mathematical statement

The supplied question asks which balanced polyhedral complexes or fans are tropical varieties of classical ideals. It does not specify the ground field, valuation, ambient space, weights, or meaning of equality. Consequently, it determines a family of realizability questions rather than one fully specified problem. The following gives a conditional formulation with these choices exposed.

Fix an integer $n\geq 1$, put $N={\mathbb{Z}}^n$, and identify $N_{{\mathbb{R}}}=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}$ with ${\mathbb{R}}^n$. In the usual pure-dimensional, weighted setting, let $\Sigma$ be a finite pure $d$-dimensional polyhedral complex with rational face directions, where $0\leq d\leq n$, and assign a positive integer weight $m_\sigma$ to each maximal cell $\sigma$. For a codimension-one face $\tau$, set \[ N_\tau=N\cap\operatorname{span}_{{\mathbb{R}}}(\tau-\tau). \] For each maximal cell $\sigma\supset\tau$, let $u_{\sigma/\tau}\in N/N_\tau$ be the primitive integral vector pointing from $\tau$ into $\sigma$ in the one-dimensional quotient direction. The weighted complex is balanced if, at every such face, \[ \sum_{\substack{\sigma\supset\tau\\ \dim\sigma=d}} m_\sigma u_{\sigma/\tau}=0 \quad\text{in }(N/N_\tau)\otimes_{{\mathbb{Z}}}{\mathbb{R}}. \] A fan is a polyhedral complex whose cells are cones with apex at the origin. The source does not explicitly impose purity, rationality, or integral weights; these are parameters of this conventional formulation, not recovered source hypotheses.

For a possible torus formulation, fix an algebraically closed field $K$ with a real-valued valuation $\nu:K^\times\to{\mathbb{R}}$, and let \[ R_K=K[x_1^{\pm1},\ldots,x_n^{\pm1}]. \] Here a classical ideal means an ordinary ideal $I\subset R_K$, as opposed to an ideal defined intrinsically over a tropical semiring. Using the minimum convention, define \[ \operatorname{Trop}_\nu(I)= \bigcap_{0\ne f\in I} \left\{w\in{\mathbb{R}}^n: \min_{a:c_a\ne0}\left(\nu(c_a)+\langle a,w\rangle\right) \text{ is attained for at least two distinct }a \right\}, \qquad f=\sum_a c_a x^a. \] For fixed $K,\nu,n,d$, a precise support version of the requested problem is to characterize all such balanced complexes $\Sigma$ for which there exists a proper ideal $I\subset R_K$, with $R_K/I$ equidimensional of Krull dimension $d$, such that \[ |\Sigma|=\operatorname{Trop}_\nu(I). \] In the weighted version, one additionally requires the prescribed weights $m_\sigma$ to agree with the tropical multiplicities induced by $I$, after a common subdivision. The analogous question applies when $\Sigma$ is a fan.

The supplied record does not decide between these versions, whether $K$ and $\nu$ are fixed or existentially quantified, whether the valuation is trivial, or whether polynomial ideals in affine space or projective ideals are intended instead. It also does not specify radicality, primality, or irreducibility. These unresolved choices must be settled before the conditional formulation above can serve as a definitive statement of the source problem.

Short English statement

Determine which balanced polyhedral complexes and fans can occur as tropical varieties of ordinary polynomial ideals. A precise version must specify the field and valuation, ambient space, and whether realization includes prescribed multiplicities or only the underlying set. The supplied record labels the problem open; this statement-only draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004214$, source problem code catalogue entry, titled Realizability of Balanced Complexes as Tropical Varieties; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2019.
• Supplied report citation: Geometric, Algebraic, and Topological Combinatorics (2020), Oberwolfach Reports, pp. 2395--2472, DOI 10.4171/owr/2019/39. https://doi.org/10.4171/owr/2019/39. The background describes the report as documenting workshop discussions and open problems; no problem-specific page or named contributor is supplied.
• Supplied literature reference, labeled "Draisma and Rincon, nonrealizable Bergman fans": https://arxiv.org/abs/1903.00356.
• Supplied literature reference, labeled "Maclagan and Rincon, tropical varieties are balanced": https://doi.org/10.1016/j.aim.2022.108713.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that concrete nonrealizability results exist but no full characterization is known. Its wording refers to balanced polyhedral complexes and tropical ideals, whereas the problem statement asks about classical ideals. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and says that notation, typography, and sentence structure were repaired while mathematical content was preserved. These are supplied editorial assertions, not mathematical verification performed for this draft.