论文标题
阿诺德的单调性问题
Arnold's monotonicity problem
论文作者
论文摘要
根据Kouchnirenko公式，具有给定牛顿多面体的通用孤立奇异性的Milnor数量等于与牛顿多面体相关的某些体积的交替总和。在本文中，我们获得了kouchnirenko公式的非负类似物（即没有负码）。该模拟依赖于Arxiv的单型操作员的非阴性公式：1405.5355和MILNOR数字的公式来自Arxiv：Math/9901107。作为一个应用程序，我们在任意维度中为Arnold的单调性问题（1982-16）提供了标准，这导致尺寸达到完整的解决方案，最高$ 4 $，dimension in Dimension $ 5 $。后者依赖于尺寸$ 2 $和$ 3 $的稀薄三角形（或消失的本地h- polynomial）的分类，来自Arxiv：1909.10843（以及Gelfand，kapranov和Zelevinsky的书），其中包含与损失的示例，这些例子与损失的例子不同，这些例子在损害上也呈现为$ 332的33232 $ 33（$ 3 $ 170 in ar arxiv：170。 ARXIV：2001.10316）。 $ 4 $维的一些示例首先在Arxiv：1309.0630中描述了局部单片猜想的背景下。
According to the Kouchnirenko formula, the Milnor number of a generic isolated singularity with given Newton polyhedron is equal to the alternating sum of certain volumes associated to the Newton polyhedron. In this paper we obtain a non-negative analogue (i.e. without negative summands) of the Kouchnirenko formula. The analogue relies on the non-negative formula for the monodromy operator from arXiv:1405.5355 and formulas for the Milnor number from arXiv:math/9901107 . As an application we give a criterion for the Arnold's monotonicity problem (1982-16) in arbitrary dimension, which leads to complete solution in dimension up to $4$ and partial solution in dimension $5$. The latter relies on the classification of thin triangulations (or vanishing local h-polynomial) in dimension $2$ and $3$ from arXiv:1909.10843 (and from the book by Gelfand, Kapranov and Zelevinsky) and contains examples which differ dramatically from the ones which arise in dimension up to $3$ in arXiv:1705.00323 (see also arXiv:2001.10316 ). Some of the $4$-dimensional examples were first described in arXiv:1309.0630 in the context of the local monodromy conjecture.
