Dimension of splines on graphs in the case of degree two and smoothness one in two variables

TL;DR

This paper investigates the dimension of degree-2 splines with smoothness one on planar graphs with specific polynomial edge labels, providing an algebraic and combinatorial approach to compute their dimension.

Contribution

It addresses the upper-bound conjecture for the dimension of these splines and introduces a graph contraction algorithm to compute the associated matrix rank.

Findings

Expresses the spline dimension in terms of the rank of an extended cycle basis matrix.

Provides a combinatorial algorithm for computing the matrix rank via graph contractions.

Addresses the upper-bound conjecture for degree-2 splines of smoothness one.

Abstract

Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data interpolation, to create smooth curves in computer graphics and to find numerical solutions to partial differential equations. Gilbert, Tymoczko, and Viel generalized the classical splines combinatorially and algebraically: a generalized spline is a vertex labeling of a graph $G$ by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the ideal generated by the corresponding edge label. We study the generalized splines on the planar graphs whose edges are labeled by two-variable polynomials of the form $(ax+by+c)^2$ and whose vertices are labeled by polynomials of degree at most two. In this paper we address the upper-bound conjecture for the dimension of degree-2 splines of smoothness 1. The dimension is expressed in terms of the rank of the extended cycle basis matrix. We also provide a combinatorial algorithm on graphs to compute the rank by contracting certain subgraphs.