# American Institute of Mathematical Sciences

June  2018, 38(6): 2827-2849. doi: 10.3934/dcds.2018120

## Introduction to tropical series and wave dynamic on them

 1 National Research University Higher School of Economics, Soyuza Pechatnikov str., 16, St. Petersburg, Russian Federation 2 IST Austria. Klosterneuburg 3400, Am campus 1, Austria

Received  June 2017 Revised  January 2018 Published  April 2018

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.

Citation: Nikita Kalinin, Mikhail Shkolnikov. Introduction to tropical series and wave dynamic on them. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2827-2849. doi: 10.3934/dcds.2018120
##### References:
 [1] E. Abakumov and E. Doubtsov, Approximation by proper holomorphic maps and tropical power series, Constructive Approximation, 47 (2018), 321-338.  doi: 10.1007/s00365-017-9375-5. [2] O. Bergman and B. Kol, String webs and $1/4$ BPS monopoles, Nuclear Phys. B, 536 (1999), 149-174. [3] E. Brugallé, Some aspects of tropical geometry, Eur. Math. Soc. Newsl., 83 (2012), 23-28. [4] E. Brugallé, I. Itenberg, G. Mikhalkin and K. Shaw, Brief introduction to tropical geometry, in Proceedings of the Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), Gökova, 2015, 1-75. [5] S. Caracciolo, G. Paoletti and A. Sportiello, Conservation laws for strings in the abelian sandpile model, EPL (Europhysics Letters), 90 (2010), 60003. [6] R. G. Halburd and N. J. Southall, Tropical Nevanlinna theory and ultradiscrete equations, Int. Math. Res. Not. IMRN, 5 (2009), 887-911. [7] N. Kalinin and M. Shkolnikov, Tropical curves in sandpiles, Comptes Rendus Mathematique, 354 (2016), 125-130.  doi: 10.1016/j.crma.2015.11.003. [8] N. Kalinin, A. Guzmán Sáenz, Y. Prieto, M. Shkolnikov, V. Kalinina and E. Lupercio, Self-organized criticality, pattern emergence, and tropical geometry, Submitted. [9] N. Kalinin and M. Shkolnikov, The number $\pi$ and summation by ${S}{L}(2, \mathbb{Z})$, Arnold Mathematical Journal, 3 (2017), 511-517, arXiv: 1701.07584. doi: 10.1007/s40598-017-0075-9. [10] N. Kalinin and M. Shkolnikov, Sandpile solitons via smoothing of superharmonic functions, Submitted, arXiv: 1711.04285. [11] N. Kalinin and M. Shkolnikov, Tropical formulae for summation over a part of $SL(2, \mathbb{Z})$, European Journal of Mathematics, arXiv: 1711.02089. [12] N. Kalinin and M. Shkolnikov, Tropical curves in sandpile models, arXiv: 1502.06284. [13] C. O. Kiselman, Croissance des fonctions plurisousharmoniques en dimension infinie, Ann. Inst. Fourier (Grenoble), 34 (1984), 155-183.  doi: 10.5802/aif.955. [14] C. O. Kiselman, Questions inspired by Mikael Passare's mathematics, Afrika Matematika, 25 (2014), 271-288.  doi: 10.1007/s13370-012-0107-5. [15] B. Kol and J. Rahmfeld, Bps spectrum of 5 dimensional field theories, (p, q) webs and curve counting, Journal of High Energy Physics, 8 (1998), Paper 6, 15 pp. [16] R. Korhonen, I. Laine and K. Tohge, Tropical Value Distribution Theory and Ultra-Discrete Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. [17] S. Lahaye, J. Komenda and J.-L. Boimond, Compositions of (max, +) automata, Discrete Event Dynamic Systems, 25 (2015), 323-344.  doi: 10.1007/s10626-014-0186-6. [18] I. Laine and K. Tohge, Tropical Nevanlinna theory and second main theorem, Proc. Lond. Math. Soc. (3), 102 (2011), 883-922.  doi: 10.1112/plms/pdq049. [19] S. Lombardy and J. Sakarovitch, Sequential?, Theoretical Computer Science, 356 (2006), 224-244.  doi: 10.1016/j.tcs.2006.01.028. [20] G. Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb R^2$, J. Amer. Math. Soc., 18 (2005), 313-377.  doi: 10.1090/S0894-0347-05-00477-7. [21] G. Mikhalkin, Tropical geometry and its applications, in International Congress of Mathematicians, Eur. Math. Soc., Zürich, 2 (2006), 827-852. [22] G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, in Curves and Abelian Varieties, vol. 465 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2008,203-230. [23] M. Shkolnikov, Tropical Curves, Convex Domains, Sandpiles and Amoebas, PhD thesis, University of Geneva, 2017. [24] K. Tohge, The order and type formulas for tropical entire functions--another flexibility of complex analysis, On Complex Analysis and its Applications to Differential and Functional Equations, 113-164. [25] T. Y. Yu, The number of vertices of a tropical curve is bounded by its area, Enseign. Math., 60 (2014), 257-271.  doi: 10.4171/LEM/60-3/4-3.

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##### References:
 [1] E. Abakumov and E. Doubtsov, Approximation by proper holomorphic maps and tropical power series, Constructive Approximation, 47 (2018), 321-338.  doi: 10.1007/s00365-017-9375-5. [2] O. Bergman and B. Kol, String webs and $1/4$ BPS monopoles, Nuclear Phys. B, 536 (1999), 149-174. [3] E. Brugallé, Some aspects of tropical geometry, Eur. Math. Soc. Newsl., 83 (2012), 23-28. [4] E. Brugallé, I. Itenberg, G. Mikhalkin and K. Shaw, Brief introduction to tropical geometry, in Proceedings of the Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), Gökova, 2015, 1-75. [5] S. Caracciolo, G. Paoletti and A. Sportiello, Conservation laws for strings in the abelian sandpile model, EPL (Europhysics Letters), 90 (2010), 60003. [6] R. G. Halburd and N. J. Southall, Tropical Nevanlinna theory and ultradiscrete equations, Int. Math. Res. Not. IMRN, 5 (2009), 887-911. [7] N. Kalinin and M. Shkolnikov, Tropical curves in sandpiles, Comptes Rendus Mathematique, 354 (2016), 125-130.  doi: 10.1016/j.crma.2015.11.003. [8] N. Kalinin, A. Guzmán Sáenz, Y. Prieto, M. Shkolnikov, V. Kalinina and E. Lupercio, Self-organized criticality, pattern emergence, and tropical geometry, Submitted. [9] N. Kalinin and M. Shkolnikov, The number $\pi$ and summation by ${S}{L}(2, \mathbb{Z})$, Arnold Mathematical Journal, 3 (2017), 511-517, arXiv: 1701.07584. doi: 10.1007/s40598-017-0075-9. [10] N. Kalinin and M. Shkolnikov, Sandpile solitons via smoothing of superharmonic functions, Submitted, arXiv: 1711.04285. [11] N. Kalinin and M. Shkolnikov, Tropical formulae for summation over a part of $SL(2, \mathbb{Z})$, European Journal of Mathematics, arXiv: 1711.02089. [12] N. Kalinin and M. Shkolnikov, Tropical curves in sandpile models, arXiv: 1502.06284. [13] C. O. Kiselman, Croissance des fonctions plurisousharmoniques en dimension infinie, Ann. Inst. Fourier (Grenoble), 34 (1984), 155-183.  doi: 10.5802/aif.955. [14] C. O. Kiselman, Questions inspired by Mikael Passare's mathematics, Afrika Matematika, 25 (2014), 271-288.  doi: 10.1007/s13370-012-0107-5. [15] B. Kol and J. Rahmfeld, Bps spectrum of 5 dimensional field theories, (p, q) webs and curve counting, Journal of High Energy Physics, 8 (1998), Paper 6, 15 pp. [16] R. Korhonen, I. Laine and K. Tohge, Tropical Value Distribution Theory and Ultra-Discrete Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. [17] S. Lahaye, J. Komenda and J.-L. Boimond, Compositions of (max, +) automata, Discrete Event Dynamic Systems, 25 (2015), 323-344.  doi: 10.1007/s10626-014-0186-6. [18] I. Laine and K. Tohge, Tropical Nevanlinna theory and second main theorem, Proc. Lond. Math. Soc. (3), 102 (2011), 883-922.  doi: 10.1112/plms/pdq049. [19] S. Lombardy and J. Sakarovitch, Sequential?, Theoretical Computer Science, 356 (2006), 224-244.  doi: 10.1016/j.tcs.2006.01.028. [20] G. Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb R^2$, J. Amer. Math. Soc., 18 (2005), 313-377.  doi: 10.1090/S0894-0347-05-00477-7. [21] G. Mikhalkin, Tropical geometry and its applications, in International Congress of Mathematicians, Eur. Math. Soc., Zürich, 2 (2006), 827-852. [22] G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, in Curves and Abelian Varieties, vol. 465 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2008,203-230. [23] M. Shkolnikov, Tropical Curves, Convex Domains, Sandpiles and Amoebas, PhD thesis, University of Geneva, 2017. [24] K. Tohge, The order and type formulas for tropical entire functions--another flexibility of complex analysis, On Complex Analysis and its Applications to Differential and Functional Equations, 113-164. [25] T. Y. Yu, The number of vertices of a tropical curve is bounded by its area, Enseign. Math., 60 (2014), 257-271.  doi: 10.4171/LEM/60-3/4-3.
The central picture shows the corner locus of the right picture which is $l_{\Omega}$ (Definition 4.1) for $\Omega = \{x^2+y^2\leq 1\}$.
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.
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.
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 $1$.
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.
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.
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.
In the picture we shrink a triangular cycle. Any deformation of a tropical curve can be decomposed into such operations or their inversions.
Computing contributions for symplectic area.
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