Affine phase retrieval and a same-matroid obstruction

Definitions and mathematical statement

Nicholas Proudfoot's Section A.15 asks about a $d$-dimensional "linear" space $V\subset{\mathbb{C}}^n$, its hyperplane-arrangement complement $M=V\cap({\mathbb{C}}^*)^n$, and \[ \operatorname{Log}:M\to\mathcal A(V),\qquad (z_1,\ldots,z_n)\mapsto(\log|z_1|,\ldots,\log|z_n|). \] The four questions are: describe fibers; give conditions making this a homeomorphism onto its image; understand amoeba topology; and relate these to Federico Ardila's matroidal description of tropicalization. A convention defect matters: if $V$ is literally a central complex linear space, multiplication by $e^{i\theta}$ preserves every Log fiber, so injectivity is impossible for nonempty positive-dimensional $M$. The source's distinct points on a complex line require an affine interpretation.

For that explicit interpretation, take an affine embedding \[ \iota:{\mathbb{C}}^d\hookrightarrow{\mathbb{C}}^n,\qquad z\mapsto(\ell_1(z),\ldots,\ell_n(z)),\qquad \ell_j(z)=a_jz+b_j, \] with no identically zero coordinate, let $H_j=\{\ell_j=0\}$, and $M={\mathbb{C}}^d\setminus\bigcup_jH_j$. Fibers are precisely the collisions of the affine intensity map $z\mapsto(|\ell_j(z)|^2)_j$ on $M$, since componentwise logarithm is injective on positive magnitudes. For $\widetilde z=(z,1)$, set $Q_j=(a_j,b_j)^*(a_j,b_j)$. Every collision $z,w\in M$ satisfies, and is characterized by, \[ H=\widetilde z\widetilde z^*-\widetilde w\widetilde w^*,\qquad \operatorname{tr}(Q_jH)=0\quad(1\leq j\leq n). \] Thus injectivity means that this measurement kernel contains no nonzero Hermitian rank-one secant difference with both points in $M$. Arbitrary rank-two Hermitian matrices are not substituted for the specified affine secants.

For an affine embedding, Log is proper onto its amoeba: a compact target set bounds every $|\ell_j|$ above and away from zero, and the embedded source then has compact preimage. Hence injectivity does imply homeomorphism onto $\mathcal A(V)$. This implication uses the embedding and finite-dimensional closed affine image, not merely arbitrary continuous injectivity.

In dimension one, write $\ell_j(z)=c_j(z-a_j)$, $c_j\ne0$. The aid's precise fiber classification is:

• Three noncollinear centers among the $a_j$ give singleton fibers.
• At least two distinct centers, all collinear, give the orbits of reflection in that real line, including fixed singleton fibers on the line.
• A single distinct center, possibly repeated, gives circles of positive radius as fibers.

This corrects the source's unqualified collinear reflection statement in the repeated-one-center case. The source's higher-dimensional construction takes products of $d$ noncollinear three-center arrangements, using $3d$ hyperplanes, and permits additional hyperplanes without losing injectivity on the remaining complement. It asks for simpler examples rather than claiming this is optimal.

The aid's same-matroid example compares centers $\{0,1,2\}$ and $\{0,1,i\}$. Their homogenized affine forms both represent the uniform matroid $U_{2,3}$, but the first has conjugation-pair fibers and the second is injective, hence a homeomorphism onto its amoeba. Consequently ordinary matroid/Bergman data alone cannot determine the Log fibers or injectivity. This does not contradict a matroidal tropicalization description in its correct linear or homogenized setting, and it is not a classification of all amoeba homotopy types.

Short English statement

The source's examples require affine, rather than central linear, spaces. In that convention Log fibers are affine magnitude-measurement collisions, and properness makes injectivity sufficient for a homeomorphism onto the amoeba. Two three-point arrangements with the same matroid have different fibers, so matroid data alone does not decide injectivity.

Sources

• Local UnsolvedMath registry, permanent ID $20001997$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[334].
• AIM workshop "Amoebas and tropical geometry," Section A.15, "Topology of amoebas of linear spaces," contributed by Nicholas Proudfoot: https://aimath.org/WWN/amoebas/amoebas.pdf. The four questions and product-of-three-center examples are retained; no PDF was opened in this session.
• Federico Ardila is named for the matroidal characterization of tropicalization, without a full authoritative bibliographic entry. Original remarks and literature fields are empty; proposer/year registry fields are null despite Proudfoot's contribution attribution.
• Supplied research attempt $1$, partial_result, run status partially_solved, rigor label "proved," novelty confidence "low." Registry classification PARTIAL-PROGRESS, updated 14 August 2026.