Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley$\unicode{x2013}$Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

翻译：广义样条是一个代数组合框架，它统一并推广了不同领域中多种已有的概念，最显著的是经典样条概念与GKM理论的拓扑概念。前者由多面体等组合几何对象上的分段多项式组成，其多项式片段在指定可微性阶数下一致；后者是一种图论构造的环面等变上同调，Shareshian与Wachs将其用于重新表述著名的Stanley–Stembridge猜想（该表述最近由Brosnan与Chow、以及Guay-Paquet独立证明成立）。本文聚焦于广义样条理论。设图$G$的每条边标记有环$R$的一个理想，图上的广义样条由$R$中元素对顶点进行标记构成，使得相邻顶点$u,v$的标记之差属于边$uv$关联的理想。我们研究广义样条的$R$-模，并给出以下若干图族与边标记族的最小生成集：1) 当所有可能的边标记集合至多包含两个有限生成理想时，对所有图成立；2) 当所有可能的边标记集合由多项式环$\mathbb{C}[x,y]$中形如$(ax+by)^2$的元素生成的主理想构成时，对循环图成立。我们通过一种适合计算机实现的构造性算法获得生成元，并给出若干应用，包括将经典（分析）样条理论中的多个结果置于上下文。