## Dimension of Multivariate Splines: An Algebraic Approach

Online event, May 31, 2021

Piecewise-polynomial functions called splines are fundamental pillars that support modern computer-aided geometric design, numerical analysis, etc. These functions are defined on polyhedral partitions of ℝ^{n}. Their restriction to any polyhedron's interior is a polynomial, and these polynomial pieces are constrained to join with some desired smoothness across hyperplanes supporting the intersections of neighboring polyhedra. Computing the dimension of spline spaces is a highly non-trivial task in general for splines in more than one variable and involves an intimate interplay of algebra, topology, and geometry. Initiated by Strang and Schumaker, this is by now a classical topic in approximation theory and has been studied in a wide range of planar settings, e.g., on T-meshes, triangulations and more polygonal meshes. Important contributions has been obtained by following and generalizing the so-called homological approach introduced in this context by Billera in 1988. The one-day online workshop focuses on the most recent developments on this topic and offers the opportunity of interaction between researchers belonging to different research areas.

#### Tchebycheffian splines over T-meshes: the homological approach

*Cesare Bracco (University of Florence, Italy)*

Tensor-product structures provide a simple and elegant construction of multivariate splines, but do not allow local refinement, which is essential for adaptive techniques both in geometric modeling and in numerical simulations. This led to the introduction of several types of adaptive spline spaces, such as T-splines, hierarchical splines, and locally refined (LR-) splines. All of them can be considered as special cases of splines over T-meshes (rectangular meshes not necessarily having the tensor-product structure). Univariate Tchebycheffian splines are smooth piecewise functions with sections in extended Tchebycheff (ET-) spaces. Thanks to the properties they share with classical polynomial splines, it is possible to define Tchebycheffian splines over T-meshes. Because of the variety of ET-spaces, such splines offer great flexibility and are then attractive for applications in geometric modeling and isogeometric analysis. The classical homological approach (introduced by Billera) to study the dimension of spline spaces has proven to be a powerful and elegant way to study spaces of splines over T-meshes and can be successfully applied to Tchebycheffian splines as well, leading to a dimension formula and giving an interesting perspective on the key features shared by polynomials and ET-spaces.

This is joint work with T. Lyche, C. Manni, F. Roman, H. Speleers.

#### Homogeneous trivariate splines on vertex stars

*Michael DiPasquale (Colorado State University, USA)*

In the 1990s Alfeld, Neamtu, and Schumaker proved a formula for the dimension of homogeneous trivariate splines on a tetrahedral vertex star continuously differentiable of order *r* and of degree *d* ≥ 3*r*+2. The formula follows the well-known pattern for splines of degree at most *d* on planar triangulations. A significant difference between the two formulas is that the dimension formula for planar triangulations is a lower bound in all degrees (Schumaker's lower bound) whereas the formula for homogeneous trivariate splines on vertex stars is not necessarily a lower bound in degrees *d* < 3*r*+2 when the vertex star is closed. We prove that the formula of Alfeld, Neamtu, and Schumaker is in fact a lower bound in degrees *d* ≥ (3*r*+2)/2 if the vertex positions are generic. Moreover, a result of Whiteley (also from the 1990s) shows that the only splines of degree *d* ≤ (3*r*+1)/2 on a generic closed vertex star are polynomials. Thus the trivial lower bound of (*d*+2)(*d*+1)/2 is the best possible (for generic vertex positions) when *d* ≤ (3*r*+1)/2, and the formula of Alfeld, Neamtu, and Schumaker is a lower bound as soon as *d* ≥ (3*r*+2)/2. The results are proved using homological and commutative algebra - particularly important is an estimation of the dimension of the vector space of polynomials of degree *k* vanishing to some order at the set of points dual to the linear forms (a so-called fat-point ideal).

This is joint work with N. Villamizar.

#### Splines arising in topology, geometry, and representation theory

*Martina Lanini (University of Rome Tor Vergata, Italy)*

In this talk I will survey how splines appear in geometry, topology and (in some lucky cases) representation theory. By presenting some examples, I will try to explain constraints, tools and questions geometers, topologists and representation theorists are concerned with while dealing with splines.

#### Algebraic tools for geometrically continuous splines

*Bernard Mourrain (Inria Sophia Antipolis Méditerranée, France)*

We study geometrically smooth spline functions that satisfy properties of differentiability on shared edges of a mesh. After having presented the context and the main ingredients associated with this type of spline functions on an arbitrary topology, we present the algebra-topological complexes associated with these spline spaces. The constructions extend the existing construction of complexes introduced by Billera for classical splines. We show how these tools help to analyze the space of geometrically continuous splines, and give information on its dimension or its bases. A few explicit examples illustrate these developments.

#### New bounds for planar splines

*Henry Schenck (Auburn University, USA)*

For a planar simplicial complex Δ contained in ℝ^{2}, Schumaker proved that a lower bound on the dimension of the space *C*^{r}_{k}(Δ) of planar splines of smoothness *r* and polynomial degree at most *k* on Δ is given by a polynomial *P*_{Δ}(*r*,*k*), and Alfeld-Schumaker showed this polynomial gives the correct dimension when *k* ≥ 4*r*+1. Examples due to Morgan-Scott, Tohaneanu, and Yuan show that the equality dim *C*^{r}_{k}(Δ) = *P*_{Δ}(*r*,*k*) can fail when *k* = 2*r* or 2*r*+1. We prove that the equality dim *C*^{r}_{k}(Δ) = *P*_{Δ}(*r*,*k*) cannot hold in general for *k* ≤ (22*r*+7)/10.

This is joint work with M. Stillman and B. Yuan.

#### Supersmoothness and dimension of multivariate splines

*Tatyana Sorokina (Towson University, USA)*

Spaces of multivariate polynomial splines are typically defined by two constants: polynomial degree (*d*) and global smoothness (*r*). Thus, the dimension formulae and formulae for bounds on dimensions depend on *d*,*r*, and the geometry of the underlying partition. We show that most (if not all) spaces of multivariate splines carry additional smoothness higher than *r* along certain faces of the partition. We show how such intrinsic additional smoothness affects the dimension and the bounds on the dimension.

#### Dimension of splines of mixed smoothness

*Deepesh Toshniwal (Delft University of Technology, The Netherlands)*

Splines on two-dimensional meshes enable the design of complex geometric objects via boundary-representations in Computer-Aided Design, as well as numerical simulations on those objects in Isogeometric Analysis. We will discuss computation of the dimension of piecewise-polynomial splines on triangular, quadrangular and polygonal meshes. The focus will be on total-degree and tensor-product splines of mixed smoothness; different orders of smoothness are chosen for different mesh elements. The utility of such splines can be motivated from an application-oriented point of view. For instance, splines with locally reduced smoothness can be used to describe surfaces of arbitrary topologies in geometric design, as well as to capture solution features of lower regularity in numerical simulations.

#### A lower bound for the dimension of tetrahedral splines in large degree

*Nelly Villamizar (Swansea University, UK)*

In the talk, we will consider splines defined on tetrahedral partitions, we derive a new formula and prove it is a lower bound on the dimension of trivariate splines in large enough degree. While this formula may fail to be a lower bound on the dimension of the spline space in low degree, we will show that in several examples this lower bound formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. We derive the bound using commutative and homological algebra.

The talk is based on a joint work with M. DiPasquale.