Ragni Piene: Algebraic splines and generalized Stanley–Reisner rings

Given a simplicial complex \(\Delta\subset \mathbb R^d\), let \(\Delta\subset \mathbb R^d\) denote the vector space of piecewise polynomial functions (algebraic splines) of degree \(\le k\) and smoothness \(r\). A major problem is to determine the dimension of these vector spaces. Pioneering work by Billera, Rose, Schenck, and others give upper and lower bounds using homological methods. 

Here we shall consider the rings  \(C^r(\Delta):=\oplus_k C^r_k(\Delta)\) that we shall call the generalized Stanley--Reisner rings of \(\Delta\). The ring of continuous splines \(C^0(\Delta)\) is (essentially) the face ring of \(\Delta\) and has the property that its geometric realization describes \(\Delta\). More precisely, the part of \({\rm Spec}(C^0(\Delta)\) lying in a certain hyperplane and having nonnegative coordinates ``is'' \(\Delta\). I propose a conjectural generalization of this situation, giving a description of \({\rm Spec}(C^r(\Delta))\) for \(r\ge 0\). To support the conjecture, some very simple examples will be given.