Exponential Sums Equations and Tropical Geometry

TL;DR

This work advances exponential-algebraic geometry by proving Exponential-Algebraic Closedness for product varieties $L\times W$ with $L$ a complex linear subspace and $W$ an algebraic subvariety of $(\mathbb{C}^\times)^n$, under additively free and rotund conditions. It develops a cohesive framework that combines amoebas, tropical geometry, and complex toric compactifications to handle both real- and complex-defined $L$, generalizing Zilber’s earlier results. The main contribution is a robust solvability result for systems of exponential sums attached to $L\times W$, which implies quasiminimality-type phenomena for exponential fields and reduces the dependence on Schanuel-type conjectures. The techniques yield a structural bridge between Newton polytopes, stable tropical intersections, and actual intersections with the graph of the exponential map, enhancing our understanding of how exponential relations behave on complex algebraic varieties. Overall, the paper broadens the geometric tools available for analyzing exponential-algebraic intersections and their model-theoretic consequences.

Abstract

We show a case of Zilber's Exponential-Algebraic Closedness Conjecture, establishing that the conjecture holds for varieties which split as the product of a linear subspace of the additive group $\mathbb{C}^n$ and an algebraic subvariety of the multiplicative group $(\mathbb{C}^\times)^n$. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required stronger assumptions on the variety such as the linear space being defined over the real numbers. The proofs use the theory of amoebas and tropical geometry.

Figures (5)

• Figure 1: The exponential map identifies the multiplicative group $\mathbb{C}^\times$ with the strip $\{z \in \mathbb{C} \mid 0 \leq \textnormal{Im}(z) \leq 2\pi \}$, glued with itself joining the outer lines: in other words, the multiplicative group is a cylinder. With this interpretation, the sets of all determinations of $1^{\sqrt{2}}$ and $1^i$ are shown in the figure: in $(a)$ we see the dense sets of points obtained as $(2\pi i \mathbb{Z})\cdot \sqrt{2} + \mathbb{Z}$, in $(b)$ the discrete set $-2\pi \mathbb{Z}$.
• Figure 2: The amoeba of the algebraic variety $W$ defined by $w_1+w_2+1=0$. We see that the amoeba has three "tentacles": the diagonal one corresponds to the behaviour of $W$ when $w_1$ and $w_2$ are both very big, and thus their absolute values are roughly the same; the vertical one corresponds to points for which $w_2$ is very close to 0 (and thus its logarithm to $-\infty$) and $w_1$ to $-1$; the horizontal one to points with $w_1$ close to 0 and $w_2$ to $-1$.
• Figure 3: This triangle is a polyhedron defined by $x_1+x_2 \leq 1$, $x_1 \geq 0$ and $x_2 \geq 0$. The three sides are faces induced by the vectors $(1,1)$, $(-1,0)$ and $(0,-1)$.
• Figure 4: (The support of) the normal fan of the polytope of Figure \ref{['triangle']}: the normal cone to the full triangle, which is the face of $(0,0)$, is the origin; the normal cones to the sides of the triangles are the three half-lines; and the normal cones to the three vertices of the triangles are the portions of space in between the half-lines. Each of the normal cones (except the origin) has been labelled to show which face of the polytope it is normal to. Considering only the origin and the three half-lines, we obtain the $1$-skeleton of the complex.
• Figure 5: The stable intersection of the usual polyhedral complex $\Sigma$ with the line $l$ is $\{0\}$: each facet intersects the line transversely. The stable intersection of $\Sigma$ with, say, the $x$-axis is still $\{0\}$: although the $x$-axis intersects the horizontal facet, the Minkowski sum of the facet and the axis does not have dimension $2$, and so the intersection does not count towards the stable intersection.

Theorems & Definitions (114)

• Conjecture 1.1: Quasiminimality Conjecture, Zil97
• Conjecture 1.2: Schanuel, see Lan66
• Theorem 1.3: BK18
• Theorem : Theorem \ref{['chap3main']}
• Definition 2.1
• Proposition 2.2: Ch
• proof
• Definition 2.3
• Theorem 2.4: Remmert's Proper Mapping Theorem, Ch
• Proposition 2.5: Ch