Stable contour graphs and their toric ends

Definitions and mathematical statement

Let $f\in{\mathbb{C}}[x_1,x_2]$ be nonzero and let $V_f=\{f=0\}\cap({\mathbb{C}}^*)^2$. On its smooth locus put \[ \operatorname{Log}(z)=(\log|z_1|,\log|z_2|),\qquad \gamma_f(z)=[z_1f_{x_1}(z):z_2f_{x_2}(z)]\in{\mathbb{P}}^1({\mathbb{C}}). \] The usual critical locus of $\operatorname{Log}|_{V_f}$ is $\gamma_f^{-1}({\mathbb{P}}^1({\mathbb{R}}))$, and its image is the amoeba contour $C_f$. This is distinct from the ramification locus, or critical points, of $\gamma_f$ itself. The source phrase "locus of the critical points of the Gauss map" is therefore retained as a definitional ambiguity and is not adopted literally as a subset of ${\mathbb{R}}^2$.

Question A.18, Thorsten Theobald. When contour singular values are taken as vertices and intervening arcs as edges, what combinatorial restrictions does the planar contour graph have, and which graphs occur for some polynomial $f$? How should the question extend to higher dimensions? The source explains that tracing the contour helps numerically determine the amoeba boundary; the contour may contain interior arcs, so it is not identified with the boundary alone. Degenerate curves and overlaps need conventions before their contours define a finite graph.

Supplied stable toric special case. Assume $V_f$ is smooth and torically nondegenerate with two-dimensional Newton polygon $\Delta$, the compactified logarithmic Gauss map is transverse to ${\mathbb{P}}^1({\mathbb{R}})$, and the contour satisfies the explicit stability/end hypotheses of the aid: finite singularities are ordinary cusps or transverse double values, and the ends are recorded individually in their normalization rather than merged solely because their limiting directions or images coincide. Let \[ B=\sum_{\text{edges }e\subset\Delta}\operatorname{length}_{{\mathbb{Z}}}(e) =\#(\partial\Delta\cap{\mathbb{Z}}^2). \] The aid asserts there are $B$ toric punctures, giving $2B$ leaves in the decorated normalized-end contour graph, paired by their punctures. Finite vertices have degree two at cusps or inserted markers and degree four at transverse double values. A marker is inserted on an otherwise vertex-free circle so that it is represented as a finite graph.

Writing $n_\times$ for the number of double values, $k$ for the number of graph components and $b_1$ for its first Betti number, the aid's exact combinatorial relation is \[ b_1=n_\times-B+k, \qquad n_\times=B-k+b_1. \] Each graph component has an even number of leaves, and finite odd-valence vertices are excluded in this class. The compactified critical normalization is asserted to be a union of circles covering ${\mathbb{P}}^1({\mathbb{R}})$; with positive covering-degree conventions their degrees sum to \[ \deg\gamma_f=2\operatorname{Area}(\Delta), \] where area gives a fundamental lattice parallelogram area one. Transversality excludes branch points lying over the real projective line. These are necessary restrictions in the specified stable toric class, not sufficiency or a classification for arbitrary polynomials; the leaf count belongs to the decorated normalized-end graph, not an arbitrary image-graph convention.

For a smooth hypersurface in $({\mathbb{C}}^*)^N$, $N>2$, one can analogously define the contour as the image of points where $\operatorname{Log}$ has rank less than $N$, equivalently the inverse image of ${\mathbb{P}}^{N-1}({\mathbb{R}})$ under the logarithmic Gauss map before taking $\operatorname{Log}$. Its generic image has dimension $N-1$ and requires a stratified-space analogue rather than a planar graph. The source does not supply a higher-dimensional realization conjecture or a complete singularity convention.

Short English statement

Theobald asks which planar graphs arise from amoeba contours and how to extend the question to higher dimensions. The supplied aid gives necessary end, valence and Euler-type restrictions under smooth toric and stability assumptions. The usual contour is formed from logarithm critical values, so the source's Gauss-critical wording and its graph conventions need explicit correction.

Sources

• Local UnsolvedMath registry, permanent ID 20002000, source code catalogue entry; canonical aim-geometry-notes.json, notes[337].
• AIM Amoebas and tropical geometry, A.18 "Contour of an amoeba", contributed by Thorsten Theobald: https://aimath.org/WWN/amoebas/amoebas.pdf.
• T. Theobald, "Computing amoebas", Exp. Math. 11 (2002), pp. 513--526; M. Passare and A. Tsikh, "Amoebas: their spines and their contours", preprint (2003), as supplied.
• Registry research attempt 1, dated 2026-08-14: partial result, rigor "proved", novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.