论文标题
热带合理功能的最小表示
Minimal Representations of Tropical Rational Functions
论文作者
论文摘要
本文研究了以下问题：鉴于分段线性函数，将其最小代数表示作为热带理性尖端。我们提出了两个不同的最小值概念，一个基于单一长度，另一个基于分解长度。我们表明，在一个维度中，这两个概念都重合，但是在两个或多个维度中并不是这样。我们证明了维数的最小表示的独特性，以及在尺寸二维中的分段线性函数的某些子类。作为证明步骤，我们在热带高度曲面排列的区域数量中获得了计数公式和下限，从而为Montúfar，ren和Zhang的结果提供了较小的扩展。作为等效的公式，它在Minkowski总和的常规混合细分中给出了顶点的下限，从而为Adiprasito的下限定理提供了Minkowski总和的较小扩展。
This paper studies the following question: given a piecewise-linear function, find its minimal algebraic representation as a tropical rational signomial. We put forward two different notions of minimality, one based on monomial length, the other based on factorization length. We show that in dimension one, both notions coincide, but this is not true in dimensions two or more. We prove uniqueness of the minimal representation for dimension one and certain subclasses of piecewise-linear functions in dimension two. As a proof step, we obtain counting formulas and lower bounds for the number of regions in an arrangement of tropical hypersurfaces, giving a small extension for a result by Montúfar, Ren and Zhang. As an equivalent formulation, it gives a lower bound on the number of vertices in a regular mixed subdivision of a Minkowski sum, giving a small extension for Adiprasito's Lower Bound Theorem for Minkowski sums.
