Discrete & Computational Geometry

, Volume 6, Issue 2, pp 107–128

A dimension series for multivariate splines

  • Louis J. Billera
  • Lauren L. Rose

DOI: 10.1007/BF02574678

Cite this article as:
Billera, L.J. & Rose, L.L. Discrete Comput Geom (1991) 6: 107. doi:10.1007/BF02574678


For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 dim Ckr(Δ)λk and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Louis J. Billera
    • 1
  • Lauren L. Rose
    • 2
  1. 1.Cornell UniversityIthacaUSA
  2. 2.Ohio State UniversityColumbusUSA