Shrinking dynamic on multidimensional tropical series

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a compact convex domain. An $\Omega$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $\Omega$ that vanishes on $\partial \Omega$. For a finite set $P\subset {\Omega }^{\circ }$, we study the least such function above prescribed initial data whose corner locus contains $P$. It is obtained by repeatedly applying one-point shrinking operators $G_p$. We prove that every fair order of these operators stabilizes after finitely many nontrivial steps. We also describe an event-driven implementation that records the lowest monomials at each point and updates only affected watcher lists. Finally, we show that, on every compact subset of ${\Omega }^{\circ }$, the resulting dynamics can be approximated by a finite path whose intermediate tropical hypersurfaces have only mild singularities on that compact set; equivalently, the corresponding local cells of the dual regular subdivision contain no lattice points other than their vertices.

arXiv commentadded proof of finiteness, and a fast allgorithm to compute the relaxation, added new pictures

Cited by in corpus (1)

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