Bounds on the dimension of trivariate spline spaces: A homological approach

摘要

We consider the vector space of globally differentiable piecewise polynomialfunctions defined on a three-dimensional polyhedral domain partitioned intotetrahedra. We prove new lower and upper bounds on the dimension of this spaceby applying homological techniques. We give an insight into different ways ofapproaching this problem by exploring its connections with the Hilbert seriesof ideals generated by powers of linear forms, fat points, the so-calledFr\"oberg-Iarrobino conjecture, and the weak Lefschetz property.