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Register a new userGeneralized splines on graphs with two labels and polynomial splines on cycles
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Added by
Jacob P. Matherne
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A generalized spline on a graph $G$ with edges labeled by ideals 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 edge-labelings consist of at most two finitely-generated ideals, and $2)$ for cycles when the edge-labelings consist 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 classical (analytic) splines.
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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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