Algebra seminar: Improving dimension bounds for trivariate splines on cells
Finding the dimension of the space of trivariate splines on cells involves the computation of a termencoding the geometry of the faces around the central vertex. The computation of such term connectsthe theory of splines with the study of fat point ideals. While the dimension of fat point ideals isquite complicated, we will show that these complications happen in low enough degree and it becomesirrelevant to bounding the dimension of the spline space. Our approach is based on results by Whiteley(1991), the reduction procedure on the fat points multiplicity introduced by Cooper, Harbourne, andTeitler (2011), and the applications of these results to estimate the Waldschmidt constant for the ideal.We will present a new combinatorial lower bound on the dimension of the spline spaces on cells, andshow that the contribution of the fat point ideal can be greatly simplified for most cells. We willapply the new bound to improve results by Colvin, DiMatteo, and Sorokina (2016), and illustrate ourapproach via examples for both generic and non-generic configurations.