Introduction to tropical series and wave dynamic on them

TL;DR

The paper develops a rigorous theory of a two-dimensional dynamic on tropical series, motivated by sandpile models and tropical-geometry connections. It defines an admissible domain $\Omega$, $\Omega$-tropical series, the weighted distance $l_{\Omega}$, and the wave operators $G_{\mathbf p}$ with the iterated dynamic $G_P$, including a lift to characteristic two; it then builds a comprehensive framework around $\mathbb{Q}$-polygons, niceness, and coarse smooth approximations to study the evolution of tropical curves. A key contribution is showing that the associated tropical symplectic area is minimized by specific constructions and that the dynamics can be effectively approximated and stabilized via blow-ups and niceness conditions, enabling a reduction to combinatorial questions about tropical curves. The work also draws connections to amoebas and the mass interpretation of webs, highlighting potential applications to sandpile dynamics and non-commutative toric varieties. Overall, the results provide a robust toolkit for analyzing tropical dynamics, offering explicit constructions, convergence properties, and geometric minimization principles that underpin further study in tropical geometry and related physical models.

Abstract

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

Figures (9)

• Figure 1: The central picture shows the corner locus of the right picture which is $l_{\Omega}$ (Definition \ref{['def_generalweighteddistance']}) for $\Omega=\{x^2+y^2\leq 1\}$.
• Figure 2: First row shows how curves given by $G_p 0_\Omega$ depend on the position of the point in the pentagon $\Omega$. The second row shows monomials in their minimal canonical form. Note that the coordinate axes of the second row are actually reversed. Each lattice point on a below picture represents a face where the corresponding monomial is dominating on a top picture, see the bottom-right picture.
• Figure 3: On the left: $\Omega$-tropical series $\min(x,y,1-x,1-y,1/3)$ and the corresponding tropical curve. On the right: the result of applying $G_{(\frac{1}{5},\frac{1}{2})}$ to the left picture. The new $\Omega$-tropical series is $\min(2x,x + \frac{2}{15},y,1-x,1-y,\frac{1}{3})$ and the corresponding tropical curve is presented on the right. The fat point is $(\frac{1}{5},\frac{1}{2})$. Note that there appears a new face where $2x$ is the dominating monomial.
• Figure 4: Illustration for Remark \ref{['rem_smooth']}. The operator $G_{\bf p}$ shrinks the face $\Phi$ where ${\bf p}$ belongs to. Firstly, $t=0$, then $t=0.5$, and finally $t=1$ in $\mathrm{Add}_{ij}^{ct}f$. Note that combinatorics of the curve can change when $t$ goes from $0$ to $1$.
• Figure 5: Examples of balancing condition in local pictures of tropical curves near vertices. The notation ${\bf m}\times (p,q)$ means that the corresponding edge has the weight $m$ and the primitive vector $(p,q).$ The vertex on the left picture is smooth, the vertices in the middle and right pictures are neither smooth nor nodal.

Theorems & Definitions (108)

• Definition 2.2
• Definition 2.3: Cf. Definition \ref{['def_tropicalanalitycal']}
• Example 2.4
• Example 2.5
• Example 2.7
• Definition 2.8
• Definition 2.9
• Proposition 2.10
• proof
• Definition 3.1