For T a planar triangulation, let R_{m} ^{r}(T) denote the space of bivariate splines on T such that f∈R_{m} ^{r}(T) is C^{r(τ)} smooth across an interior edge τ and, for triangle σ in T, f|_{σ} is a polynomial of total degree at
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For T a planar triangulation, let R_{m} ^{r}(T) denote the space of bivariate splines on T such that f∈R_{m} ^{r}(T) is C^{r(τ)} smooth across an interior edge τ and, for triangle σ in T, f|_{σ} is a polynomial of total degree at most m(σ)∈Z_{≥0}. The map m:σ↦Z_{≥0} is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of R_{m} ^{r}(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of R_{m} ^{r}(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in R_{m} ^{r}(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013).

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