We use basic facts in tropical intersection theory [13] to adapt the geometrical presentation of tropical elliptic curve group law from [4] to the case of the tropical Hesse curve. Let $A\subset \{i,j,k\in \mathbb{N}:i+j+k=d\}$ be called the *support* of the polynomial

$f(x,y,z)=\sum _{(i,j,k)\in A}{a}_{ijk}{x}^{i}{y}^{j}{z}^{k}$

defining a plane tropical curve. The convex hull of points $(i,j,k,{a}_{ijk})$ is a 3-dimensional polytope, the lower faces of which project bijectively, under omission of the last coordinate, onto the planar convex hull of *A*, thus defining a *regular subdivision* Δ of *A*. The segments of ${V}_{f}$ arise from the interior edges of Δ, and the tentacles arise from its boundary edges [16]. Figure 2, right, shows the regular subdivision of the tropical Hesse curve. The *weight* of a facet of the tropical curve is the lattice length of its dual edge in the regular subdivision(i.e. the number of lattice points on the edge, including extremities, minus 1). The *degree* of the curve is *d* as defined in the support of its polynomial.

Then the balancing condition holds: for any vertex *V* in the tropical curve ${V}_{f}$ with adjacent edges ${E}_{1},\mathrm{\dots},{E}_{p}$, let ${w}_{i},{v}_{i}$ be the weight and unit vectors of edge ${E}_{i}$; then ${w}_{1}{v}_{1}+\mathrm{\dots}+{w}_{n}{v}_{n}=0$, where $0=(0,0)$ of ${\mathbb{P}}^{2}(\mathbb{T})$. For a vertex having exactly three adjacent edges, the *multiplicity* is defined as the common quantity

${w}_{2}{w}_{3}|det({v}_{2},{v}_{3})|={w}_{3}{w}_{1}|det({v}_{3},{v}_{1})|={w}_{1}{w}_{2}|det({v}_{1},{v}_{2})|$

and following [4] a tropical curve is called *smooth* if all its vertices are 3-valent and have multiplicity 1, and a *tropical elliptic curve* is a smooth tropical curve of degree 3 and genus 1 (number of cycles). More generally, the intersection of two segments of a plane tropical curves, with respective weight ${w}_{1},{w}_{2}$ and unit vectors ${v}_{1},{v}_{2}$, has multiplicity ${w}_{1}{w}_{2}|det({v}_{1},{v}_{2})|$ (see [16]).

Inspection of Figure 2 (right) show that the weight of the Hesse curve’s tentacles are 3, and the weight of the bounded segments are 1 and the balancing condition checks for all three vertices. The degree is 3 and its genus is 1. All vertices are 3-valent, their multiplicity, however, is e.g. for the origin *O*:

${m}_{O}=1\times 1\times \left|\left|\begin{array}{cc}\hfill -2\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill -2\hfill \end{array}\right|\right|=3$

and the tropical Hesse curve is not smooth. The cycle ${C}_{K}$ of the tropical curve ${H}_{K}$ is obtained simply by removing the three tentacles, its equation is given by $\mathrm{max}(3X,3Y,0)=K+X+Y$.

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