The Newton Polytope
The Newton Polytope
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牛顿多面体
DOI：
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发表时间：
2006
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通讯作者：
R. Edman
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作者：
R. Edman
这以精确的顺序描述了映射χ。序列中的映射L给出了一个矩阵，使得ImL = kerχ。这个矩阵将生成一个整数格Λ <$Zm-n。这样一个确切的顺序引起了许多不同的解释。最熟悉的解释是R中的多面体，它将几何引入系统。我们也可以把精确序列看作定义一个微分方程组。多面体的体积将与微分方程的解的数目有关。由于矩阵χ的列生成一个整数格，因此引入一种体积形式，使基本区域的体积为整数值，这对我们将是有用的。对于任意整数格Λ <$Z，我们考虑所有顶点在Λ中的n维单形，一个初等单形将是Λ中具有最小体积的n维单形。Λ上的体积形式将给出初等单形体积1。同样，该体积形式给出
This describes the map χ in the exact sequence. The map L in the sequence gives us a matrix such that ImL = kerχ. This matrix will generate a integer lattice Λ ⊆ Zm−n. Such an exact sequence gives rise to many different interpretations. The most familiar interpretation is that of a polytope in R which introduces geometry to the system. We can also view the exact sequence as defining a system of differential equations. The volume of the polytope will be related to the number of solutions to the differential equations. Since the columns of the matrix χ generate a integer lattice, it will be useful to us to introduce a volume form which makes the volume of fundamental regions integral values. For any integer lattice Λ ⊂ Z we consider all the n dimensional simplices with vertices in Λ, an elementary simplex will be an n simplex in Λ with minimal volume. The volume form on Λ will give elementary simplices volume 1. Equivalently, this volume form give