Shrinking dynamic on multidimensional tropical series

Abstract

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