This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of a prescribed order of smoothness on some subdivided domain D in R^k, and appear in applications ranging from approximation theory to geometric modeling to numerical analysis. Alternatively, splines can be interpreted as a collection of polynomials labeling the vertices of a (combinatorial) graph, with adjacent vertex-labels differing by a power of an affine linear form attached to the edge. In most cases of interest, the subdivided domain is essentially dual to the combinatorial graph, and these two characterizations of splines coincide. Properties of splines depend on combinatorics, topology, geometry, and symmetry of a simplicial or polyhedral subdivision of a region D in R^k, and are often quite subtle. We describe how duality, group actions, and homology -- techniques which play a central role in many areas of both pure and applied mathematics -- can be used to illuminate different questions about splines. Our target audience is nonspecialists: we provide a concrete introduction to these methods, and illustrate them with many examples in the context of splines. We also provide a tutorial on computational aspects: all of the objects appearing in this note may be studied using open source computer algebra software.

翻译：本文综述了与样条研究相关的三个核心代数主题：对偶性、群作用与同调。样条是定义在R^k中某个细分区域D上具有指定光滑阶数的分段多项式函数，其应用涵盖逼近理论、几何建模和数值分析等领域。此外，样条也可解释为标记在（组合）图顶点上的多项式集合，其中相邻顶点标签之差等于附着在边上某个仿射线性形式的幂次。在多数重要情形下，细分区域本质上与该组合图对偶，且样条的这两种刻画方式相互等价。样条的性质依赖于R^k中区域D的单纯或多面体细分的组合结构、拓扑性质、几何特征及对称性，往往颇为微妙。本文阐述了对偶性、群作用与同调——这些在纯数学与应用数学众多领域均发挥核心作用的技术——如何被用于阐释关于样条的不同问题。面向非专业读者，我们提供这些方法的具体导引，并通过样条背景下的诸多实例加以说明。同时，我们给出计算方面的教程：本文涉及的所有对象均可利用开源计算机代数软件进行研究。