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Algebraic splines and generalized Stanley-Reisner rings

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Algebraic geometry; Computer science; Applied computer science

video/mp4

Author affiliation: University of Oslo

2017-02-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60564

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/