Fat points, inverse systems, and piecewise polynomial functions (Q1270988)

The authors review and explore recently discovered, interesting new connections between the three topics mentioned in the title of the paper. In J. Algebra 174, No. 3, 1080-1090 (1995; Zbl 0842.14002), \textit{J. Emsalem} and \textit{A. Iarrobino} observed that there is a relationship between fat points in \(\mathbb{P}^n\) (i.e. zero-dimensional subschemes \(\mathbb{X}\) defined by ideals of the form \(I_{\mathbb{X}}= {\mathfrak p}_1^{\alpha_1} \cap\cdots\cap {\mathfrak p}_s^{\alpha_s}\)) and ideals generated by powers of linear forms. More precisely, for \(j\geq \max\{\alpha_1,\dots, \alpha_s\}\), the \(j\)-th graded piece of the Macaulay inverse system \((I_{\mathbb{X}})^{-1}\) equals the corresponding graded piece of a certain ideal generated by \(s\) powers of linear forms. In the case of linearly independent linear forms \(\{L_1,\dots, L_s\}\) in two indeterminates, the authors compute explicit formulas for the Hilbert function of \(J= (L_1^{\alpha_1},\dots, L_s^{\alpha_s})\), for the socle degree of \(k[y_0,y_1]/J\), and thus for the minimal graded free resolution of \(J\). In the paper: Compos. Math. 108, No. 3, 319-356 (1997; Zbl 0899.13016), \textit{A. Iarrobino} also observed that there is a relationship between splines (i.e. piecewise polynomial functions satisfying certain smoothness conditions) on a \(d\)-dimensional simplicial complex \(\Delta\) and the ideals generated by powers of the linear forms defining hyperplanes incident to the interior faces of \(\Delta\). The authors recall a certain chain complex whose top homology module is precisely the module \(C^\alpha (\widehat{\Delta})\) of mixed splines on the cone \(\widehat{\Delta}\) over \(\Delta\) which are smooth of order \(\alpha\). In the case of planar splines \(\Delta\subset \mathbb{R}^2\), they are then able to provide formulas for the number of splines in \(C^\alpha (\widehat{\Delta})_k\) of sufficiently large degrees \(k\gg 0\). The paper ends with a discussion of the obstacles which have to be overcome in order to get higher-dimensional versions of those results.