This thesis concerns the algebra $C^r(\PC)$ of $C^r$ piecewise polynomial functions (splines) over a subdivision by convex polytopes $\PC$ of a domain $\Omega\subset\R^n$.Interest in this algebra arises in a wide variety of contexts, ranging from approximation theory and computer-aided geometric design to equivariant cohomology and GKM theory.A primary goal in approximation theory is to construct bases of the vector space $C^r_d(\PC)$ of splines of degree at most $d$ on $\PC$, although even computing the dimension of this space proves to be challenging.From the perspective of GKM theory it is more important to have a good description of the generators of $C^r(\PC)$ as an algebra; one would especially like to know the multiplication table for these generators (the case $r=0$ is of particular interest).For certain choices of $\PC$ and $r$ there are beautiful answers to these questions, but in most cases the answers are still out of reach.In the late 1980s Billera formulated an approach to spline theory using the tools of commutative algebra, homological algebra, and algebraic geometry~\cite{Homology}, but focused primarily on the simplicial case.This thesis details a number of results that can be obtained using this algebraic perspective, particularly for splines over subdivisions by convex polytopes.The first three chapters of the thesis are devoted to introducing splines and providing some background material.In Chapter~\ref{ch:Introduction} we give a brief history of spline theory.In Chapter~\ref{ch:CommutativeAlgebra} we record results from commutative algebra which we will use, mostly without proof.In Chapter~\ref{ch:SplinePreliminaries} we set up the algebraic approach to spline theory, along with our choice of notation which differs slightly from the literature.In Chapter~\ref{ch:Continuous} we investigate the algebraic structure of continous splines over a central polytopal complex (equivalently a fan) in $\R^3$.We give an example of such a fan where the link of the central vertex is homeomorphic to a $2$-ball, and yet the $C^0$ splines on this fan are not free as an algebra over the underlying polynomial ring in three variables, providing a negative answer to a question of Schenck~\cite[Question~3.3]{Chow}.This is interesting for several reasons.First, this is very different behavior from the case of simplicial fans, where the ring of continuous splines is always free if the link of the central vertex is homeomorphic to a disk.Second, from the perspective of GKM theory and toric geometry, it means that the multiplication tables of generators will be much more complicated.In the remainder of the chapter we investigate criteria that may be used to detect freeness of continuous splines (or lack thereof).From the perspective of approximation theory, it is important to have a basis for the vector space $C^r_d(\PC)$ of splines of degree at most $d$ which is `locally supported' in some reasonable sense.For simplicial complexes, such a basis consists of splines which are supported on the union of simplices surrounding a single vertex.Such bases are well known in the case of planar triangulations for $d\ge 3r+2$~\cite{HongDong,SuperSpline}.In Chapter~\ref{ch:LSSplines} we show that there is an analogue of locally-supported bases over polyhedral partitions, in the sense that, for $d\gg 0$, there is a basis for $C^r_d(\PC)$ consisting of splines which are supported on certain `local' sub-partitions.A homological approach is particularly useful for describing what these sub-partitions must look like; we call them `lattice complexes' due to their connection with the intersection lattice of a certain hyperplane arrangement.These build on work of Rose \cite{r1,r2} on dual graphs.It is well-known that the dimension of the vector space $C^r_d(\PC)$ agrees with a polynomial in $d$ for $d\gg 0$.In commutative algebra this polynomial is in fact the Hilbert polynomial of the graded algebra $C^r(\wPC)$ of splines on the cone $\wPC$ over $\PC$.In Chapter~\ref{ch:AssPrimes} we provide computations for Hilbert polynomials of the algebra $C^\alpha(\Sigma)$ of mixed splines over a fan $\Sigma\subset\R^3$, giving an extension of the computations in~\cite{FatPoints,TMcD,TSchenck09,Chow}.We also give a description of the fourth coefficient of the Hilbert polynomial of $HP(C^\alpha(\Sigma))$ where $\Sigma=\wDelta$ is the cone over a simplicial complex $\Delta\subset\R^3$.We use this to re-derive a result of Alfeld-Schumaker-Whiteley on the generic dimension of $C^1$ tetrahedral splines for $d\gg 0$~\cite{ASWTet} and indicate via example how this description may be used to give the fourth coefficient in particular non-generic configurations.These computations are possible via a careful analysis of associated primes of the spline complex $\cR/\cJ$ introduced by Schenck-Stillman in~\cite{LCoho} as a refinement of a complex first introduced by Billera~\cite{Homology}.Once the Hilbert polynomials which give the dimension of the spaces $C^r_d(\PC)$ for $d\gg 0$ are known, one would like to know how large $d$ must be in order for this polynomial to give the correct dimension of the vector space $C^r_d(\PC)$.Indeed the formulas are useless in practice without knowing when they give the correct answer.In the case of a planar triangulation, Hong and Ibrahim-Schumaker have shown that if $d\ge 3r+2$ then the Hilbert polynomial of $C^r(\wPC)$ gives the correct dimension of $C^r_d(\PC)$~\cite{HongDong,SuperSpline}.In the language of commutative algebra and algebraic geometry, this question is equivalent to asking about the \textit{Castelnuovo-Mumford regularity} of the graded algebra $C^r(\wPC)$.In Chapter~\ref{ch:Regularity}, we provide bounds on the regularity of the algebra $C^\alpha(\Sigma)$ of mixed splines over a polyhedral fan $\Sigma\subset\R^3$.Our bounds recover the $3r+2$ bound in the simplicial case.The proof of these bounds rests on the homological flexibility of regularity, similar in philosophy to the Gruson-Lazarsfeld-Peskine theorem bounding the regularity of curves in projective space (see~\cite[Chapter 5]{Syz}).
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