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Algebraic splines and generalized Stanley-Reisner rings Piene, Ragni


Given a simplicial complex \(\Delta\subset \mathbb R^d\), let \(C^r_k(\Delta)\) 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 (and construct bases) of these vector spaces. Pioneering work by Billera, Rose, Schenck, and others gave upper and lower bounds using homological methods. The ring of continuous splines \(C^0(\Delta)=C^0_k(\Delta)\) is (essentially) equal to the face ring, or Stanley--Reisner 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 ``equal'' to \(\Delta\). Here we shall consider the \emph{generalized Stanley--Reisner rings} \(C^r(\Delta):=\oplus_k C^r_k(\Delta)\subset C^0(\Delta)\). We present a conjectural description of \({\rm Spec}(C^r(\Delta))\) generalizing the one for \(r=0\). To illustrate the conjecture, some very simple examples will be given.

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