Title:Shrinking dynamic on multidimensional tropical series

Abstract:We define multidimensional tropical series, i.e. piecewise linear function which are tropical polynomials locally but may have infinite number of monomials. Tropical series appeared in the study of the growth of pluriharmonic functions. However our motivation originated in sandpile models where certain wave dynamic governs the behaviour of sand and exhibits a power law (so far only experimental evidence). In this paper we lay background for tropical series and corresponding tropical analytical hypersurfaces in the multidimensional setting. The main object of study is $\Omega$-tropical series where $\Omega$ is a compact convex domain which can be thought of the region of convergence of such a series.
Our main theorem is that the sandpile dynamic producing an $\Omega$-tropical analytical hypersurface passing through a given finite number of points can always be slightly perturbed such that the intermediate $\Omega$-tropical analytical hypersurfaces have only mild singularities.