Abstract
For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceC rk (Δ) consisting of allC r piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑ k≥0 dimℝ C r k (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)=f d (Δ), andP′(1)=(r+1)f 0d−1 (Δ). We discuss how the polynomialP(λ) and bases for the spacesC r k (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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Alfeld, On the dimension of multivariate piecewise polynomial functions, inProceedings of the Biennial Dundee Conference on Numerical Analysis, Pitman, London, 1985.
P. Alfeld, B. Piper, and L. L. Schumaker, An explicit basis forC 1 quartic bivariate splines,SIAM J. Numer. Anal. 24 (1987), 891–911.
P. Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1, Constr. Approx. 3 (1987), 189–197.
M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.
D. A. Bayer, The Division Algorithm and the Hilbert Scheme, Ph.D. Dissertation, Harvard University, 1982.
L. J. Billera, The algebra of continuous piecewise polynomials,Adv. in Math. 76 (1989), 170–183.
L. J. Billera, Homology of smooth splines: generic triangulations and a conjecture of Strang,Trans. Amer. Math. Soc. 310 (1988) 325–340.
L. J. Billera and L. L. Rose,Gröbner basis methods for multivariate splines, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. L. Schumaker, eds., Academic Press, New York, 1989, pp. 93–104.
B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, inMultidimensional Systems Theory, N. K. Bose, ed., Reidel, Dordrecht, 1985, pp. 184–232.
C. K. Chui and R. H. Wang, On smooth multivariate spline functions,Math. Comp. 41 (1983), 131–142.
H. Crapo and J. Ryan, Spatial realizations of linear scenes,Structural Topology 13 (1986), 33–68.
B. Grünbaum,Convex Polytopes, Wiley-Interscience, London, 1967.
R. Hartshorne,Algebraic Geometry, Spring-Verlag, New York, 1977.
I. Kaplansky,Commutative Rings, University of Chicago Press, Chicago, IL, 1974.
D. Lazard, Gröbner bases, Gaussian elimination, and resolution of systems of algebraic equations, inComputer Algebra, Proceedings EU ROCAL '83, J. A. van Hulzen, ed., Lecture Notes in Computer Science, Vol. 162, Springer-Verlag, New York, 1983, pp. 146–156.
F. S. Macaulay, Some properties of enumeration in the theory of modular systems,Proc. London Math. Soc. 26 (1927), 531–555.
H. Matsumura,Commutative Algebra, Second Edition, Benjamin, London, 1980.
P. Orlik,Introduction to Arrangements, CBMS Regional Research Conference Series, No. 72, American Mathematical Society, Providence, RI, 1989.
L. Robbiano, Term orderings on the polynomial ring,Proceedings of EUROCAL 85, Lecture Notes in Computer Science, Vol. 204, Springer-Verlag, New York, 1985, pp. 513–517.
L. L. Rose, The Structure of Modules of Splines over Polynomial Rings, Ph.D. Thesis, Cornell University, Ithaca NY, January, 1988.
L. L. Rose and H. Terao, Hilbert polynomials and geometric lattices,Adv. in Math. (to appear).
K. Saito, Theory of logarithmic differential forms and logarithmic vector fields,J. Fac. Sci. Univ. Tokyo Sect. IA 27 (1980), 265–291.
F.O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstraßschen Divisionsatz und eine Anwendung auf analytische Cohen-Macaulay, Stellenalgebren minimaler Multiplizität, Diplomarbeit am Fachbereich Mathematik der Universitat Hamburg, 1980.
L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement,Adv. in Math. 64 (1987), 305–325.
D. A. Spear, A constructive approach to commutative ring theory,Proceedings of the 1977 MACSYMA Users' Conference, NASA CP-2012, National Technical Information Service, Springfield, VA, 1977, pp. 369–376.
R. P. Stanley,Combinatorics and Commutative Algebra, Birkhauser, Boston, 1983.
R. P. Stanley,Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.
P. F. Stiller, Certain reflexive sheaves onP nC and a problem in approximation theory,Trans. Amer. Math. Soc. 279 (1983), 125–142.
P. F. Stiller, Vector bundles on complex projective spaces and systems of partial differential equations, I,Trans. Amer. Math. Soc. 298 (1986), 537–548.
B. Sturmfels and N. White, Gröbner bases and invariant theory,Adv. in Math. 76 (1989), 245–259.
H. Terao, Arrangements of hyperplanes and their freeness, I,J. Fac. Sci. Univ. Tokyo Sect. IA 27 (1980), 293–312.
W. Whiteley, Realizability of polyhedra,Structural Topology 1 (1979), 46–58.
O. Zariski and P. Samuel,Commutative Algebra, Vol. II, Van Nostrand, Princeton, NJ, 1960; Springer-Verlag, New York, 1975.
G. M. Ziegler, Algebraic Combinatorics of Hyperplane Arrangements, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, May, 1987.
G. M. Ziegler, Combinatorial construction of logarithmic differential forms,Adv. in Math. 76 (1989), 116–154.
G. M. Ziegler, Multiarrangements of hyperplanes and their freeness, inProceedings, International Conference on Singularities, Iowa City, Iowa, 1986, R. Randell, ed., Contemporary Mathematics, Vol. 90, American Mathematical Society, Providence, RI, 1989.
Author information
Authors and Affiliations
Additional information
This research was partially supported by NSF Grants DMS-8403225 and DMS-8703370/DMS-8896193.
Rights and permissions
About this article
Cite this article
Billera, L.J., Rose, L.L. A dimension series for multivariate splines. Discrete Comput Geom 6, 107–128 (1991). https://doi.org/10.1007/BF02574678
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574678