Lattice-supported splines on polytopal complexes

We study the module Cr(P)Cr(P) of piecewise polynomial functions of smoothness r on a pure n  -dimensional polytopal complex P⊂RnP⊂Rn, via an analysis of certain subcomplexes PWPW obtained from the intersection lattice of the interior codimension one faces of PP. We obtain two main results: first, we show that in sufficiently high degree, the vector space Ckr(P) of splines of degree ⩽k   has a basis consisting of splines supported on the PWPW for k≫0k≫0. We call such splines lattice-supported  . This shows that an analog of the notion of a star-supported basis for Ckr(Δ) studied by Alfeld–Schumaker in the simplicial case holds [3]. Second, we provide a pair of conjectures, one involving lattice-supported splines, bounding how large k   must be so that dimRCkr(P) agrees with the McDonald–Schenck formula [14]. A family of examples shows that the latter conjecture is tight. The proposed bounds generalize known and conjectured bounds in the simplicial case.