Subalgebras of the Stanley-Reisner ring (Q1293673)

Let \(\Delta\) be a simplicial complex on the vertex set \(\{v_1,\dots,v_n\}\). The Stanley-Reisner ring of \(\Delta\) is defined to be \(A_{\Delta}={\mathbb R}[Y_1,\dots,Y_n]/I_{\Delta}\) where \(I_{\Delta}\) is the ideal of non-faces of \(\Delta\). Denote by \(C^r(\Delta)\) the algebra of piecewise polynomial functions on \(\Delta\) of smoothness \(r\) (\(r\)-splines), and by \(C^r_k(\Delta)\) the space of splines of smoothness \(r\), for which each polynomial is of degree at most \(k\). \textit{L. J. Billera} [Adv. Math. 76, No. 2, 170-183 (1989; Zbl 0703.13015)] proved that \(C^0(\Delta)\) is isomorphic as an \({\mathbb R}\)-algebra to \(A_{\Delta}/\sum_{i=1}^n Y_i-1\). In this paper a criterion is given to determine which elements of the Stanley-Reisner ring correspond to splines of higher-order smoothness. Lau and Stiller pointed out that the dimension of \(C^r_k(\Delta)\) is upper semicontinuous in the Zariski topology. Using the criterion, the author gives an algorithm for obtaining the defining equations of the set of vertex locations where the dimension jumps.