Algebraic Structure of Generalized Splines

Abstract

Given a fnite graph G=(V,E), a commutative ring R with unity and an edge labeling function α that assigns the ideals of R to the edges of G, the pair (G,α) is called an edge labeled graph. A vertex labeling F ⊂ R |V | is said to be a generalized spline on (G,α) if the difference of the labels on adjacent vertices is an element of the ideal on the corresponding edge. The collection of all generalized splines on (G,α) over the base ring R is denoted by R(G,α) . There exists a ring and an R-module structure on R(G,α) . The module structure is studied with number of methods such as fow-up bases, the Chinese Remainder Theorem and linear algebra techniques in this thesis. We focus on the problems of freeness and finding bases for generalized spline modules. We give a combinatorial method to find the smallest leading entries of flow-up classes on any graph over principal ideal domains. We introduce a basis criteria for generalized spline modules on cycle graphs, diamond graphs and trees over greatest common divisor domains by using some determinantal techniques. We define the homogenization of an edge labeled graph to get more information about the generalized spline modules. This thesis includes six chapters. We give a survey of the literature on classical and generalized spline theory in Chapter 1. We give a detailed movitation of generalized spline theory. We summarize the results of the thesis. In Chapter 2, we give the necessary background knowledge such as the properties of CGD and LCM, the Chinese Remainder Theorem and the fundamentals of module theory and graph theory. In Chapter 3, we introduce basic definitions and properties of generalized spline theory. We study algebraic properties of the set R(G,α) and investigate the effect of changing the ordering of the vertices of (G,α) on the module structure of R(G,α) . Also, we define the matrix M(G,α) which is used for finding R-module generators of R(G,α) . In Chapter 4, we focus on a specific type of generalized splines, which is called flow-up classes. We formulate the smallest leading entries of flow-up classes on any graphs over any principal ideal domains by using some combinatorial techniques. We also investigate the existence of flow-up bases for R (G,α) . Moreover, we give an algorithm to compute flow-up classes that have smallest leading entries on arbitrary ordered cycles. In Chapter 5, we give a basis criteria for generalized spline modules on cycles, dia- mond graphs and trees over greatest common divisor domains by using determinantal techniques and flow-up classes. We generalize some previous works which are done for cycles and diamond graphs over integers and we introduce a basis criteria for generalized spline modules on trees. In order to do this, we use the smallest leading entries of flow-up classes that we formulate in Chapter 4. In Chapter 6, we defne the homogenization of an edge labeled graph in order to give a graded module structure to the set of generalized splines. We study the freeness relation between R(G,α) and the module of its homogenization. We also give some applications of the basis criteria that we introduce in Chapter 5.