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Abstract
For a pure polyhedral complex of dimension d, embedded in ${\rm I\!R}\sp{\rm d}$, we consider C$\sp{\rm r}(\Delta$), the ${\rm I\!R}$-algebra of piecewise polynomial functions on $\Delta$ which are continuously differentiable of order r. These functions are sometimes called splines or finite elements and are useful in a variety of applications. If R is the polynomial ring over ${\rm I\!R}$ in d variables, then R embeds in C$\sp{\rm r}(\Delta$) as the global polynomial functions on $\Delta$. In this way C$\sp{\rm r}(\Delta$) becomes an R-module. The aim of this thesis is to study the structure of C$\sp{\rm r}(\Delta$) as an R-module.
First we study the question of freeness for C$\sp{\rm r}(\Delta$) over R. When $\Delta$ is a simplicial complex, we obtain a local criterion for freeness. In this case we completely characterize freeness for C$\sp{0}(\Delta)$, extending a result of Billera. In particular, we show that freeness for C$\sp0(\Delta$) depends only on the combinatorial properties of $\Delta$. For arbitrary $\Delta$, we find, in general, that freeness is only combinatorial when d is equal to 1 or 2.
Let C$\sbsp{\rm k}{\rm r}(\Delta$) denote the ${\rm I\!R}$-subspace of C$\sp{\rm r}(\Delta$) consisting of elements of degree less than or equal to k. We show that the series $\sigma$ dim$\sb{\rm I\!R}$C$\sbsp{\rm k}{\rm r}(\Delta)\lambda\sp{\rm k}$ has the form P(A$\lambda$)/(1-$\lambda)\sp{\rm d+1}$ where P($\lambda$) is in $\doubz\lbrack\lambda$), and we determine some combinatorial invariants of this function. Moreover, we see that for sufficiently large k, the function D(k) = dim$\sb{\rm I\!R}$C$\sbsp{\rm k}{\rm r}(\Delta$) is given by a polynomial of degree d in ${\rm I\!Q}$(k).
When C$\sp{\rm r}(\Delta$) is a free R-module, we establish a criterion for C$\sp{\rm r}(\Delta$) to have a "reduced" basis, a basis from which we can construct bases for the C$\sbsp{\rm k}{\rm r}(\Delta)\sp\prime$s over ${\rm I\!R}$. When $\Delta$ is a hereditary complex, in particular when $\Delta$ is a manifold, we describe methods for computing these bases.
