二维判别式coamoebas
Discriminant coamoebas in dimension two
摘要：
本文讨论某些特殊平面代数曲线在坐标辐角映射下的象。这些曲线是两个变量的简化A-判别式的零集。我们首先把coamoeba看作环面T^2=（R/2πZ）^2的子集，但也把它看作其覆盖空间R^2的子集，在这种情况下，coamoeba由一个无限的双周期映象组成。事实上，考虑到多重性是很自然的，因此在代数拓扑学的意义上，我们可以把coamoeba当作一条链。我们给出了一个非常明确的描述coamoeba作为工会的两个镜像（一般非凸）多边形，这是很容易从矩阵B，表示盖尔变换的原始集合A。我们还给出了一个面积公式的coamoeba，我们表明，coamoeba是密切相关的一定zonotope。事实上，在环面T^2上，变形虫和环带体一起形成了一个环，因此精确地覆盖了整个环面的整数倍。证明了这个整数等于A的凸包的（归一化）体积。
英文摘要：
This paper deals with coamoebas, that is, images under coordinatewise argument mappings, of certain quite particular plane algebraic curves. These curves are the zero sets of reduced A-discriminants of two variables. We consider the coamoeba primarily as a subset of the torus T^2=(R/2πZ)^2, but also as a subset of its covering space R^2, in which case the coamoeba consists of an infinite, doubly periodic image. In fact, it turns out to be natural to take multiplicities into account, and thus to treat the coamoeba as a chain in the sense of algebraic topology. We give a very explicit description of the coamoeba as the union of two mirror images of a (generally non-convex) polygon, which is easily constructed from a matrix B that represents the Gale transform of the original collection A. We also give an area formula for the coamoeba, and we show that the coamoeba is intimately related to a certain zonotope. In fact, on the torus T^2 the coamoeba and the zonotope together form a cycle, and hence precisely cover the entire torus an integer number of times. This integer is proved to be equal to the (normalized) volume of the convex hull of A.
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