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Introduction to tropical series and wave dynamic on them

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  • 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.

    Mathematics Subject Classification: Primary: 14T05, 11S82, 37J05, 11H06; Secondary: 37E15, 37K60, 82C20, 35J05, 37P50.

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  • Figure 1.  The central picture shows the corner locus of the right picture which is $l_{\Omega}$ (Definition 4.1) 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 2. 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 ${\text{Add}}_{ij}^{ct}f$. 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 $(p,q).$ From left to right: a smooth vertex, a nodal vertex, then two neither smooth nor nodal vertices.

    Figure 6.  Left: the curve corresponding to the function $g$ from Theorem 9.5, near a corner, $d(S_k) = 4$. Each vertex $V$ of the curve is smooth because $g$ is locally presented as $\min(y,kx,(k+1)x)$ near $V$. Right: an example of $C(g)$ for $g$ in Lemma 12.3. Colored corners symbolize that a quasidegree was not nice, and we made blow-ups at these corners.

    Figure 7.  Above pictures show non-unimodular corners $\Lambda$ (dashed lines). The corresponding below pictures present lattice points with respect to whom we should perform the blow-ups in Lemma 12.3, in order to make all the corners unimodular: the result is shown by continuous lines above. Dashed lines below show vectors dual to the new sides.

    Figure 8.  In the picture we shrink a triangular cycle. Any deformation of a tropical curve can be decomposed into such operations or their inversions.

    Figure 9.  Computing contributions for symplectic area.

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