样条曲线mod m
Splines mod m
摘要：
给定一个图，其边由环中的理想标记，广义样条是每个顶点由环元素标记，使得邻顶点相差与该边相关联的理想的元素。研究环Z/mZ上的样条函数。以前的工作认为样条域，其中发生非常不同的现象。例如，当环是整数时，样条模的基的元素由图的顶点索引。然而，我们证明了在Z/mZ样条模本质上可以有1和n之间的任何秩。利用有限Z-模的分类，我们开始了Z/mZ上样条的分类工作，并产生了Z/mZ上循环样条的最小生成集。我们以许多开放的问题结束。
英文摘要：
Given a graph whose edges are labeled by ideals in a ring, a generalized spline is a labeling of each vertex by a ring element so that adjacent vertices differ by an element of the ideal associated to the edge. We study splines over the ring Z/mZ. Previous work considered splines over domains, in which very different phenomena occur. For instance when the ring is the integers, the elements of bases for spline modules are indexed by the vertices of the graph. However we prove that over Z/mZ spline modules can essentially have any rank between 1 and n. Using the classification of finite Z-modules, we begin the work of classifying splines over Z/mZ and produce minimum generating sets for splines on cycles over Z/mZ. We close with many open questions.
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