Associated Primes of Spline Complexes

The spline complex $\mathcal{R}/\mathcal{J}[\Sigma]$ whose top homology is the algebra $C^\alpha(\Sigma)$ of mixed splines over the fan $\Sigma\subset\mathbb{R}^{n+1}$ was introduced by Schenck-Stillman in [Schenck-Stillman 97] as a variant of a complex $\mathcal{R}/\mathcal{I}[\Sigma]$ of Billera [Billera 88]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of $C^\alpha(\Sigma)$, including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of $\Sigma$, generalizing the computations in [Geramita-Schenck 98,McDonald-Schenck 09]. The second is a description of the fourth coefficient of the Hilbert polynomial of $HP(C^\alpha(\Sigma))$ for simplicial fans $\Sigma$. We use this to derive the result of Alfeld, Schumaker, and Whiteley on the generic dimension of $C^1$ tetrahedral splines for $d\gg 0$ [Alfeld-Schumaker-Whiteley 93] and indicate via an example how this description may be used to give the fourth coefficient in particular nongeneric configurations.Comment: 40 pages, 10 figure

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems

A Lower Bound for Splines on Tetrahedral Vertex Stars

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly diffcult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra|we call theseclosed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of splines whose derivatives up to order r are continuous on closed vertex stars of degree at least 3r + 2. We show that this formula is a lower bound on the dimension of splines of degree at least (3r + 2)/2. Our proof uses apolarity and thevso-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most (3r + 1)/2 on a generic closed vertex star are global polynomials

Splines on polytopal complexes

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})