Piecewise polynomials on polyhedral complexes

Terry McDonald, Hal Schenck

Research output: Contribution to journalArticlepeer-review


For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f (P, r, k) of degree d. When d = 2 and P is simplicial, Alfeld and Schumaker give a formula for all three coefficients of f. However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension-2 linear spaces, we obtain the first three coefficients of f (P, r, k), giving a complete answer when d = 2.

Original languageEnglish (US)
Pages (from-to)82-93
Number of pages12
JournalAdvances in Applied Mathematics
Issue number1
StatePublished - Jan 2009


  • Dimension formula
  • Hilbert polynomial
  • Polyhedral spline

ASJC Scopus subject areas

  • Applied Mathematics


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