Algorithms for Amoeba Dimensions from Tropical Fans

Research-attempt provenance

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A rational polyhedral cone is generated by finitely many rational vectors with nonnegative real coefficients. A rational polyhedral fan is a finite collection of such cones closed under taking faces, with intersections common faces; its support $F$ is their union. Its dimension is the maximum cone dimension. A rational subspace $U\subseteq\mathbb R^n$ is the real span of rational vectors. The Minkowski sum is $U+F=\{u+f:u\in U,f\in F\}$, whose dimension is the maximum dimension of its polyhedral pieces. The input is a finite rational description of the fan; the output is the minimum displayed in the source question.

Statement recorded in the supplied source.

Given a $d$-dimensional rational polyhedral fan with support $F\subseteq\mathbb R^n$, compute $\min_U\{2\dim_{\mathbb R}(U+F)-\dim_{\mathbb R}U\}$ over subspaces $U$ spanned by rational vectors. [S2]

Scope qualification.

The minimization allows every rational subspace, including zero and the whole ambient space. The question asks for an algorithm; no polynomial runtime or restriction to coordinate subspaces is added.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Given a rational tropical fan, compute the best dimension expression obtained by adding a rational subspace.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005478 (OWR-12697711-014). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 New Directions in Real Algebraic Geometry $2023$, Oberwolfach Reports, DOI 10.4171/owr/2023/15, pp. 813--881 https://doi.org/10.4171/owr/2023/15.
• S3 Draisma--Eggleston--Pendavingh--Rau--Yuen, Proceedings of the AMS https://doi.org/10.1090/proc/16744. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30005478 and source code OWR-12697711-014 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Literature and context check

I read Draisma's primary report contribution on pages 855--857, especially Open question 1 on page 856. It asks for an algorithm on arbitrary rational fans; the polynomial-time matroidal result later in the talk is a different restricted setting. The following reduction and example are elementary auxiliary work and do not assume that every rational fan is the tropicalization of a linear space.

Research attempt: Rank reduction, a three-cone obstruction, and an uncertified limit search

Replacing cone geometry by rational linear algebra.

Let the nonempty finite rational fan have cones $\sigma_1,\ldots,\sigma_s$, and put $L_i=\operatorname{span}\sigma_i$. For a rational subspace $U$, \[ \dim(U+F)=\max_i\dim(U+L_i). \] Indeed $U+F$ is the finite union of $U+\sigma_i$. Each such polyhedral set has linear span $U+L_i$ and nonempty relative interior there: a relative interior point of the cone, added to $U$, has the expected quotient-relative neighborhood. Equivalently projecting the cone to $(U+L_i)/U$ preserves a full-dimensional relative interior in that quotient. Its dimension is therefore $\dim(U+L_i)$, and the dimension of a finite union is the maximum. All these dimensions are exact rational matrix ranks. If $d=\max_i\dim L_i$, the objective is at least $d$. To see this, write $k=\dim U$ and choose a cone of dimension $d$; then $2\dim(U+L_i)-k\geq2\max(k,d)-k\geq d$. Let $L=\sum_iL_i$ and $r=\dim L$. Restricting to $U\subseteq L$ loses nothing: put $U_0=U\cap L$ and $e=\dim U-\dim U_0$. Since $U\cap L_i=U_0\cap L_i$, $\dim(U+L_i)=e+\dim(U_0+L_i)$; the objective decreases by exactly $e$ when $U$ is replaced by $U_0$. Taking $U=L$ gives the upper bound $r$, and taking $U=0$ gives $2d$. For a pure fan with every maximal cone span of dimension $d$, taking their common intersection $C$ gives $2d-\dim C$.

An exact case defeating two natural candidate choices.

In $\mathbb R^6$, take the three coordinate cones \[ \sigma_1=\operatorname{cone}(e_1,e_2,e_4),\quad \sigma_2=\operatorname{cone}(e_2,e_3,e_5),\quad \sigma_3=\operatorname{cone}(e_1,e_3,e_6), \] with all their faces. Their pairwise intersections are coordinate rays, which are faces of each cone, so this really is a rational polyhedral fan. Its maximal spans have dimension three, common intersection zero, and total span $\mathbb R^6$. Both $U=0$ and $U=\mathbb R^6$, and also the common intersection choice, give value six. But $U=\operatorname{span}(e_1,e_2,e_3)$ meets each maximal span in dimension two, hence each sum has dimension four and the objective is five. This is optimal. For an arbitrary $k$-dimensional $U$, write $t=\min_i\dim(U\cap L_i)$. The value is $k+6-2t$. For $k=0$ it is six; for $k=1$ it is at least five; for $k\geq5$ it is at least $k\geq5$. For $k=2$, a value below five would require $t=2$, forcing $U$ into the zero common intersection, impossible. For $k=3$, a value below five would require $t=3$, forcing all three $L_i$ to equal $U$, impossible. For $k=4$, a value below five would again require $t=3$, forcing their six-dimensional sum into $U$, impossible. Thus the minimum is exactly five. Only testing common intersections and the whole span is not a correct algorithm for the original optimization.

A convergent enumeration is not a terminating algorithm.

Enumerate all rational matrices with at most $r$ columns, for example by increasing bounds on numerator and positive denominator sizes, and take their column spans as candidates $U\subseteq L$. Maintain the smallest exact rank objective seen, starting from $r$. Its values are integers between $d$ and $r$. The original minimum is attained by some rational subspace, since the set of attained integer values is a nonempty finite subset of $\{d,\ldots,r\}$. A rational basis for an optimal subspace eventually appears in the enumeration, so the maintained bound eventually equals the answer and stays there. However, finitely many changes in an integer sequence do not give an effective way to recognize its last change. Hitting the lower bound $d$ certifies termination in that special case; in the displayed example the answer is $5>d=3$, so that stopping rule cannot apply. A proof of a finite candidate family or an effective exclusion test for smaller values is still missing.

Verification and adversarial audit

The example uses pointed coordinate cones with all faces, rather than merely listing subspaces that would fail the fan intersection axiom. Faces of smaller dimension do not increase the maximum rank. The exact optimum proof checks all $k=0,\ldots,6$. The search distinguishes eventual stabilization from a computable stopping test and does not invoke real quantifier elimination as though it automatically decided rational-subspace existence.

Audit corrections.

Corrected the geometric example to a genuine fan of coordinate cones and faces. Separated rational-subspace enumeration from decidability and avoided importing the source's special matroid result into the arbitrary-fan setting.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The rank reduction, exact example and convergent upper-bound search are substantive but auxiliary. No always-terminating algorithm for arbitrary rational fans is proved. In particular the rational witness question and a finite stopping certificate remain, so the original algorithmic target is unresolved.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/47009 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.