A finite recognition certificate for planar tropical fans

Definitions and mathematical statement

The original Section C.1, "Relevant lines of research," is a thematic list rather than a quantified proposition. It lists: basic definitions; computational issues; higher-codimension amoebas; example families; abstract data types in tropical and idempotent calculus; recognition problems; applications to complex algebraic geometry, real algebraic geometry, dynamical systems, differential equations, optimization and control, representation theory, number theory and statistical mechanics; and tropical representation theory. No domain, desired output or theorem is attached to these headings.

The aid supplies a separate precise recognition problem inspired by that list. In ${\mathbb{R}}^2$, take distinct rays with primitive integer generators $u_1,\ldots,u_m\in{\mathbb{Z}}^2\setminus\{0\}$, in cyclic order, and independently specified positive integer tropical cell weights $w_1,\ldots,w_m$. Suppose the directions span ${\mathbb{R}}^2$. The reported criterion is \[ \text{the weighted rays form the codimension-one skeleton of the normal fan of a lattice polygon} \quad\Longleftrightarrow\quad \sum_{i=1}^m w_i u_i=0. \] Here the normal fan uses outward normals and the max-plus convention; the weight of a ray is the lattice length of its dual polygon edge. Equivalently this is a constant-coefficient one-vertex planar tropical hypersurface.

Let $J(x,y)=(-y,x)$. For accepted full-rank data the explicit polygon is reconstructed by \[ p_0=0,\qquad p_k=\sum_{i=1}^k w_iJu_i\quad(1\leq k\leq m), \qquad p_m=p_0. \] Its edge directions, lengths and convex cyclic order give the unique Newton polygon up to translation. A corresponding tropical polynomial can use its vertices with equal coefficients. The finite certificate checks integer primitivity, distinct rays, cyclic order, rank two, positive integer weights, and the two-coordinate balance sum, followed by these exact prefix sums.

For rank-one nonempty balanced distinct primitive-ray data, there are exactly two antipodal rays $u,-u$, with the same weight $w>0$. They form the multiplicity-$w$ binomial line, dual to the lattice segment $[0,wJu]$. This degenerate case is not represented as a full-dimensional polygon.

Normalization conventions matter. If input uses a nonprimitive representative $v_i=d_i u_i$, $d_i>0$, and $w_i$ independently denotes the tropical cell weight, replace $v_i$ by $u_i$ while retaining $w_i$. The different rule $w_i\mapsto d_iw_i$ is appropriate only when the raw product $w_iv_i$ was explicitly intended as the balancing vector. Confusing these encodings changes the recognized tropical cycle. The criterion is a reported special synthesis; it is not a theorem that every source research direction has been solved, nor recognition for arbitrary nonconstant-coefficient tropical varieties.

Short English statement

The source is a list of research directions. Its aid develops one recognition example: primitive weighted planar rays give a lattice-polygon normal fan exactly when they balance, with an explicit polygon reconstruction. Rank-one data give a weighted binomial line, and ray normalization must preserve the declared cell-weight convention.

Sources

• Local UnsolvedMath registry, permanent ID $20002006$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[343], zero-based source index $343$, tag "section."
• AIM workshop "Amoebas and tropical geometry," 23--26 October 2003, Section C.1, "Relevant lines of research": https://aimath.org/WWN/amoebas/amoebas.pdf. No PDF was opened in this drafting session.
• Supplied research attempt $1$, result context_only, run status not_a_problem, rigor label "proved," novelty confidence "low." Registry status "open," classification OPEN-TRIAGE, suggested difficulty L2, updated 14 August 2026. The recognition theorem is the aid's mathematical synthesis, not an original question stated in C.1.
• Original remarks and literature fields are empty; no proposer, proposed year or authoritative external citation for the recognition criterion is supplied.