A family of ideals of minimal regularity and the Hilbert series of Cr(Δ)

Hal Schenck, Mike Stillman

Research output: Contribution to journalArticlepeer-review


For a simplicial subdivision Δ of a region inR2, we analyze the dimension of the vector spaceCkr(Δ) ofCrpiecewise polynomial functions (splines) on Δ of degree at mostk. We find an exact sequence which allows us to prove that the dimension series for splines given by5does indeed agree with the bounds on the dimension of the spline space given by Alfeld and Schumaker [1;2]. We give sufficient conditions for the Alfeld-Schumaker bounds to be attained in all degrees, where Δ is a two-dimensional simplicial complex. The conditions are satisfied by the class of complexes considered by6, but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, this result is the strongest possible.

Original languageEnglish (US)
Pages (from-to)169-182
Number of pages14
JournalAdvances in Applied Mathematics
Issue number2
StatePublished - Aug 1997
Externally publishedYes

ASJC Scopus subject areas

  • Applied Mathematics


Dive into the research topics of 'A family of ideals of minimal regularity and the Hilbert series of Cr(Δ)'. Together they form a unique fingerprint.

Cite this