A multivariate spline is a piecewise polynomial function on a partition of some domain which is continuously differentiable to order for some integer . Multivariate splines play an important role in many areas such as finite elements, computer-aided design, and data fitting [20, 11]. In these applications it is important to construct a basis, often with prescribed properties, for splines of bounded total degree. A more basic task which aids in the construction of a basis is simply to compute the dimension of the space of multivariate splines of bounded degree on a fixed partition. We write
for the vector space of piecewise polynomial functions of degree at moston the partition which are continuously differentiable of order .
A formula for the dimension of , where is a planar triangulation, was first proposed by Strang  and proved for by Billera . Subsequently the problem of computing the dimension of planar splines on triangulations has received considerable attention using a wide variety of techniques, see [31, 2, 3, 19, 36, 37, 5, 6, 29, 28]. Ibrahim and Schumaker show in  that the dimension of , for a planar triangulation and , is given by a quadratic polynomial in whose coefficients are determined from simple data of the triangulation. An important feature of planar splines is that the formula which gives the dimension of the spline space for is a lower bound for any degree .
In this paper we focus on splines over the union of tetrahedra all of which meet at a common vertex. We call such a configuration a star of a vertex (these are sometimes called cells in the approximation theory literature [20, 32]). If is the star of a vertex, every spline can be written as a sum of homogeneous splines; a homogeneous spline of degree is one which restricts to a homogeneous polynomial of degree on each tetrahedron. We denote by the vector space of homogeneous splines of degree in . Understanding homogeneous splines on vertex stars is crucial to computing the dimension of trivariate splines on tetrahedral complexes (see ) – whose behavior even in large degree is a major open problem in numerical analysis. We apply our present results on vertex stars to tetrahedral splines of large degree in a forthcoming paper.
In , Alfeld, Neamtu, and Schumaker derive formulas for the dimension of the space of homogeneous splines on vertex stars of degree . A crucial difference from the planar case is that when these formulas may not even be a lower bound on the dimension of the space of homogeneous splines. To explain this we differentiate between two types of vertex stars. If the common vertex at which all tetrahedra meet is completely surrounded by tetrahedra (so that it is the unique interior vertex), then we call the vertex star a closed vertex star. Otherwise we call the vertex star an open vertex star. In Equations 15 and 16 of , Alfeld, Neamtu, and Schumaker define functions (in terms of simple geometric and combinatorial data of ) which we denote by (11) and (12), respectively. In [1, Theorem 3], it is shown that for if is a closed vertex star and that for if is an open vertex star.
It is straightforward to show that for all if is an open vertex star. On the other hand, if is a closed vertex star it is quite delicate to determine the degrees for which ; see  where a lower bound is established for homogeneous splines on vertex stars. Our main result is a simple bound on the degrees for which is a lower bound on .
Theorem 1.2 (Lower bound for splines on vertex stars).
If is a closed vertex star with interior vertex and interior edges, put
If then and
The failure of to be a lower bound for in low degree is elucidated by homological techniques of Billera  as refined by Schenck and Stillman . More precisely, it follows from these techniques (in particular the Billera-Schenck-Stillman chain complex) that
where is an ideal generated by powers of linear forms attached to the interior vertex (see Proposition 4.4). Iarrabino showed that, via apolarity, can be computed from the Hilbert function of a so-called ideal of fat points in . The Hilbert function of an ideal of fat points in is the subject of much research (and a major open conjecture) in algebraic geometry [26, Section 5]. Fortunately, we need relatively little information about the Hilbert function of this ideal of fat points to establish Theorem 1.2 – a sufficiently good lower bound on the so-called Waldschmidt constant [34, 7] of the dual set of points is enough to establish that for . Evidently the inequality (2) then implies if .
Next we turn to the question of small degree, namely when . If we use the inequality (2), finding entails some difficult fat point computations. However, (2) is often not the best possible lower bound for in small degree. In fact, it is often easier to analyze directly in small degree, bypassing the difficulty of computing entirely. Whiteley  completed just such an analysis for generic planar triangulations; we prove the following variation on his result for closed vertex stars.
Theorem 1.3 (Low degree splines on generic closed vertex stars).
If is a generic closed vertex star with interior vertex then for .
See Definition 2.7 for the meaning of a generic vertex star.
Theorem 1.3 shows that, at least for generic vertex positions, the best lower bound in degrees is also the simplest. Thus, if vertex positions are generic, one cannot obtain a ‘better’ lower bound by computing for . We do not know when it is possible to bypass the computation of in low degree for non-generic vertex positions, although we show the general strategy for one non-generic configuration in Section 6.2. Our result suggests that, just as in the planar case, the main difficulty in computing in low degree is understaning the non-trivial homology module of the Billera-Schenck-Stillman chain complex.
The paper is organized as follows. In Section 2 we set up notation and briefly describe the homological machinery from [5, 29]. In Section 3 we use apolarity and the Waldschmidt constant to show that if (see the above discussion). In Section 4 we prove Theorem 1.2, and in Section 5 we prove Theorem 1.3. Section 6 is devoted to illustrating our bounds in some examples and Section 7 contains concluding remarks.
2. Background and Homological Methods
In this section we review the necessary results from  and . We denote by a simplicial complex embedded in (see  for basics on simplicial complexes). If we will refer to as a triangulation, and as a tetrahedral complex if . We denote by the set of interior faces of of dimension , and by the number of such faces for . If we call an -face. By an abuse of notation, we will identify with its underlying space .
Recall that a simplicial complex is said to be pure if all maximal simplices have the same dimension. A pure -dimensional simplicial complex is hereditary if, whenever two maximal simplices intersect in a vertex , then there is a sequence of -dimensional simplices satisfying that for and for .
If is a pure -dimensional simplicial complex all of whose -dimensional simplices share a common vertex then we call the star of a vertex or a vertex star. Without loss of generality, we assume that is at the origin. If is an interior vertex of then we call a closed vertex star; if is on the boundary of then we call an open vertex star. The link of a pure -dimensional vertex star in which all -dimensional simplices share the vertex is the set of all simplices of which do not contain (this has dimension ). Throughout this article, whenever we refer to a simplicial complex , we will mean a pure, hereditary, -dimensional vertex star whose link is simply connected. We call these tetrahedral vertex stars, simplicial vertex stars, or simply vertex stars.
We write for the polynomial ring in three variables, for the vector space of homogeneous polynomials of degree , and for the vectors space of polynomials of total degree at most . For a given integer , we denote by the set of all functions which are continuously differentiable of order .
Let be a tetrahedral vertex star. The space of splines on is the piecewise polynomial functions on that are continuously differentiable up to order on i.e.,
|If we consider polynomials of degree at most , the space will be denoted by , namely|
|Similarly, the space of splines whose polynomial pieces are of degree exactly is defined as|
The space is itself a ring, and includes naturally into as global polynomials. In this way is both an -module and a -algebra. We will be concerned exclusively with the structure of as an -vector space; however we may at times refer to the -module structure of . In particular, if is the star of a vertex, then it is known that
where the isomorphism is as -vector spaces. If and , notice that . This means that has the structure of a graded -module.
If has more than one interior vertex, there is a coning construction under which (3) will still be valid. As we focus on the case of vertex stars, we will not need this.
Suppose is an tetrahedral vertex star. If , let be a choice of linear form vanishing on . We define , the ideal generated by . For any face where we define
If we define .
[6, Proposition 1.2] If is hereditary then if and only if
We define a chain complex introduced by Billera  and refined by Schenck and Stillman . We refer to  for undefined terms from algebraic topology. We denote the simplicial chain complex of relative to its boundary with coefficients in by :
The ideals fit together to make a sub-chain complex of (the differential is the restriction of the differential of ):
The Billera-Schenck-Stillman chain complex is the quotient of by , namely
These three chain complexes fit into the evident short exact sequence of chain complexes
If is a tetrahedral vertex star whose link is simply connected, then , for , and .
The inclusion of into as globally polynomial corresponds (via the isomorphism in Proposition 2.5) to the copy of in , while the map
encodes the so-called smoothing cofactors. By Proposition 2.5, the Billera-Schenck-Stillman chain complex and the chain complex contain essentially the same information.
We now put everything together to write the dimension of in terms of the Billera-Schenck-Stillman chain complex. If is a chain complex of graded -modules, we write for the graded Euler-Poincaré characteristic of . That is,
If is a tetrahedral vertex star then
We use the fact that . If is a closed vertex star with interior vertex , then has the form
the map is surjective from the definition of , hence . Thus
The result follows since by Proposition 2.5.
If is an open vertex star, then has the form
so there is not even a vector space in homological index , thus as well and the formula follows from the above argument immediately. ∎
Finally, we clarify what we mean by generic vertex positions for a tetrahedral vertex star; this is fairly standard in the literature on splines [37, 36, 5, 4, 1]. The main point is that it will suffice to prove Theorem 1.2 when is a generic tetrahedral vertex star.
Suppose is a star of the vertex . We call generic with respect to a fixed if, for all sufficiently small perturbations of the vertices , the resulting vertex star satisfies . If and are understood from context, then we simply say is generic.
Suppose is a star of the vertex , and fix non-negative integers and . Then almost all sufficiently small perturbations of the vertices result in a vertex star which is generic with respect to and . Moreover .
This follows immediately from examining rank conditions on any of the equivalent ways of defining splines as the kernel of a linear transformation. ∎
3. Duality: fat points and powers of linear forms
In this section we review a duality between ideals generated by powers of linear forms and ideals of polynomials which vanish to certain orders on sets of points in projective space, called fat point ideals. We reduce the presentation to our case i.e., ideals in the polynomial ring of three variables and the corresponding fat points ideals in . We use this duality, along with combinatorial bounds from , to prove Corollary 3.18, the main result of this section. Corollary 3.18 provides explicit lower bounds for the degree in which , where is the interior vertex of a closed vertex star.
We write for a point in projective -space over , which we denote by . We let be the polynomial ring in three variables. If we write for the ideal of homogeneous polynomials in which vanish at ; i.e. . It is straightforward to see that consists of all polynomials whose homogeneous components vanish to order at .
Let be a collection of points in and a collection of positive integers attached to , respectively. The ideal of fat points associated to and is
If there is a positive integer so that for then we write instead of . If for we simply write .
It is straightforward to see that is the set of polynomials whose homogeneous components vanish to order at the point , for . Since is graded for each , is also a graded ideal.
The ideal in Definition 3.1 is called the th symbolic power of , and consists of the polynomials whose homogeneous components vanish to order on . The th symbolic power can be defined for any ideal , but the definition given above for points is all we will need.
Now consider simultaneously the polynomial rings and , and let act on as polynomial differential operators. Namely, if and then . We call this the apolarity action of on . The apolarity action induces a perfect pairing by .
Let . If , then .
If is an ideal of , then the inverse system of is defined as
If is graded, then is a graded vector space (it is generally not an ideal) with graded structure , where is the vector space of all homogeneous polynomials of degree in .
Let , with . Then . More generally, if then .
For a graded ideal , the apolarity action induces an isomorphism of vector spaces (this follows since the apolarity action induces a perfect pairing ). Thus one can deduce from , and vice-versa. The following result of Iarrobino  describes the inverse system of a fat point ideal.
We first introduce some notation which suits our context. If is a two-simplex in whose affine span contains the origin, let be a choice of linear form vanishing on (well-defined up to constant multiple). The coefficients of define the point (notice this point does not depend on the multiple of chosen), which in turn defines the ideal .
Theorem 3.6 (Iarrabino ).
Let be a collection of -simplices in each of whose affine span contains the origin and let be a positive integer attached to each . Put . If , let and . Then
Theorem 3.6 has an especially nice formulation in the case of uniform powers. We state this for the ideal of the interior vertex of a closed vertex star, as this is the case of interest to us.
Suppose is a vertex star with unique interior vertex , so . Put and let . Then
Let be the regular octahedron with central vertex at the origin and vertices at and . Then there are 12 interior two-dimensional faces which we denoted as ; we number them so that they lie in the planes defined by the linear forms , for , for , and for . The dual points are , , and , see the graph on the left of Figure 1. These points define the ideals , and . For a positive integer , and let . Theorem 3.6 says that
For example, if and then and . On the other hand, and .
Let now be a generic octahedron with central vertex at the origin. Then the 12 two-dimensional faces lie on 12 different planes through the origin of defined by the linear forms . These linear forms define 12 points in . Notice that each of the edges lies in the intersection of four of these planes, in that means that the four dual points to the linear forms vanishing at those four planes lie on a line – thus there is a dotted line in Figure 1 for every interior edge of . The dual diagram in is illustrated on the right in Figure 1.
3.1. The Waldschmidt constant
Given a graded ideal , we put . For instance, if is a set of points in , then is the minimum degree of a homogeneous polynomial which vanishes on . An asymptotic invariant attached to the ideal which has been studied in many different contexts is the Waldschmidt constant, defined as
It is known that the Waldschmidt constant is actually a limit (this follows from subbaditivity of the sequence - see [7, Lemma 2.3.1]); so .
The limit was first introduced by Waldschmidt  in complex analysis, although the ideas behind the Waldschmidt constant go back at least to Nagata’s solution to Hilbert’s fourteenth problem. In commutative algebra, the Waldschmidt constant gives bounds related to the containment problem; in other words for what pairs of integers we have the containment for an ideal in a polynomial ring (the first use of the Waldschmidt constant for these purposes is in ).
For a closed vertex star , let be a finite subset of two-faces such that . Put , , and . Then for .
By Corollary 3.7, if and only if . We may assume (otherwise ). Dividing both sides by gives if and only if
Since the right hand side is larger than , we see that
Solving for yields the proposition, provided that . This latter inequality follows from a result of Chudnovsky that (see [17, Proposition 3.1]). Thus if then , so if (and only if) , that is is contained in a line. But this would imply that the span of the corresponding linear forms is at most two dimensional, contrary to assumption. ∎
3.2. A reduction procedure for fat points
Following the notation introduced in Section 3, let denote a collection of faces in of a vertex star . The dual points defined by the linear forms vanishing on these faces define the dual points . Consider a collection of non-negative integers , and the fat points ideal .
If , then is the intersection of at two (distinct) planes in which contain the faces having as one of their edges. Since , then the dual points lie in a common line in . By construction, for each interior edge , the corresponding dual line contains at least two points for .
In the following we describe the procedure introduced by Cooper, Harbourne, and Tietler in  to give bounds on . This is done by constructing the so-called reduction vector, which we now describe. Given a sequence of non-negative integers , a collection of points , and the sequence of lines of not necessarily different lines from the collection , the vector is defined inductively as follows.
Starting with , we define as the number of points lying on , counted with multiplicity. Namely, if for some , then .
Reduce by 1 the multiplicities of all the points lying on and consider the sequence of points now with multiplicities for , and for .
A reduction vector is said to be a full reduction vector for the fat points ideal if .
Let be the regular octahedron from Example 3.8. Taking for every , and the ideal of fat points . The set of lines in this case is . If we take , two of the points lie on , each of them with multiplicity 2, so . Notice that . See Figure 2, where we produce a reduction vector following starting from the dual graph of in .
Let be the generic octahedron from Example 3.9. Let us take for every , and the ideal of fat points given by . To construct a reduction vector associated to the ideal , we can take any sequence of lines in , in particular we can take a sequence so that all multiplicities reduce to zero. For instance, following the notation in Figure 3, by taking the sequence of lines the multiplicity at each point is reduced to 2. If we continue the reduction following the sequence of lines in the same order one more time, we get , and all the multiplicities are reduced to zero.
In  it is shown that the reduction vector yields bounds on . In the statement of the following theorem (and throughout this document) it is important that we use the convention that if .
[10, Corollary 2.1.5] Let be a fat points ideal and a full reduction vector from the sequence of lines . Let and for let
If the reduction vector does not contain zeros (it is positive), the following corollary to Theorem 3.14 provides bound on the initial degree of the ideal of fat points . (Crucially, for reading the following theorem, the indexing of the reduction vector was reversed between the preprint  and its publication ).
[9, Theorem 4.2.2] If is an ideal of fat points in which has a positive full reduction vector , then
Suppose is a closed vertex star with interior vertex and the two-faces of all span distinct planes. Let be the set of points dual to the collection of forms . Then .
Since the two-faces of all span distinct planes, the set