Advanced Search
Article Contents
Article Contents

Introduction to tropical series and wave dynamic on them

Abstract Full Text(HTML) Figure(9) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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.

  • [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. LahayeJ. 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.
  • 加载中



Article Metrics

HTML views(356) PDF downloads(190) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint