多维热带级数上的收缩动力学
Shrinking dynamic on multidimensional tropical series
AI总结：
本文研究紧凸域上热带级数的收缩动力学，证明单点收缩算子的公平顺序有限步稳定，并给出事件驱动实现及近似路径的奇点性质。
摘要：
设 $\Omega\subset\mathbb R^n$ 为紧凸域。$\Omega$-热带级数是定义在 $\Omega$ 上、非负、凹、整斜率、分段仿射且在边界 $\partial\Omega$ 上取零值的函数。对于有限集 $P\subset\Omega^\circ$，我们研究在给定初始数据之上、其角点轨迹包含 $P$ 的最小此类函数。该函数通过反复应用单点收缩算子 $G_p$ 获得。我们证明，这些算子的每个公平顺序在有限个非平凡步骤后稳定。我们还描述了一种事件驱动的实现，该实现记录每个点的最低单项式，并仅更新受影响的观察者列表。最后，我们证明，在 $\Omega^\circ$ 的每个紧子集上，所得动力学可被一条有限路径近似，该路径的中间热带超曲面在该紧集上仅具有轻微奇点；等价地，对应的对偶正则细分中的局部胞腔除顶点外不含格点。
英文摘要：
Let $Ω\subset\mathbb R^n$ be a compact convex domain. An $Ω$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ$. For a finite set $P\subsetΩ^\circ$, 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 $Ω^\circ$, 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.
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