Mathematics in Computer Science

, Volume 8, Issue 2, pp 157–174 | Cite as

Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach



We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight of different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Fröberg–Iarrobino conjecture, and the weak Lefschetz property.


Triangulations Splines Dimension Tetrahedral partition Ideals of powers of linear forms Fröberg’s conjecture 

Mathematics Subject Classification

13 14 68 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alfeld P., Schumaker L.L.: Bounds on the dimensions of trivariate spline spaces. Adv. Comput. Math. 29(4), 315–335 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alfeld P., Schumaker L.L., Sirvent M.: On dimension and existence of local bases for multivariate spline spaces. J. Approx Theory 70(2), 243–264 (1992)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Alfeld P., Schumaker L.L., Whiteley W.: The generic dimension of the space of C 1 splines of degree d ≥ 8 on tetrahedral decompositions. SIAM J. Numer. Anal. 30(3), 889–920 (1993)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Anick D.J.: Thin algebras of embedding dimension three. J. Algebra 100(1), 235–259 (1986)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Billera L.J.: Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Am. Math. Soc. 310(1), 325–340 (1988)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Brambilla, M.C., Dumitrescu, O., Postinghel, E.: On a notion of speciality of linear systems in \({\mathbb{P}^n}\). Trans. Am. Math. Soc (2013) arXiv:1210.5175v2 [math.AG]
  7. 7.
    Chandler K.A.: The geometric interpretation of Fröberg–Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space. J. Algebra 286(2), 421–455 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ciliberto, C.: Geometric Aspects of Polynomial Interpolation in More Variables and of Waring’s Problem, European Congress of Mathematics, Vol. I (Barcelona, 2000), pp. 289–316 (2001)Google Scholar
  9. 9.
    Cottrell J.A., Hughes T.J.R., Bazilevs Y.: Isogeometric analysis: Toward integration of CAD and FEA. Wiley, New York (2009)CrossRefGoogle Scholar
  10. 10.
    Eisenbud, D., Grayson, D., Stillman, M.: Macaulay2, Software system for research in algebraic geometry.
  11. 11.
    Fröberg R.: An inequality for Hilbert series of graded algebras. Math. Scand 56(2), 117–144 (1985)MATHMathSciNetGoogle Scholar
  12. 12.
    Geramita A.V., Schenck H.: Fat points, inverse systems, and piecewise polynomial functions. J. Algebra 204(1), 116–128 (1998)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    A.V. Geramita: Inverse Systems of Fat Points: Waring’s Problem, Secant Varieties of Veronese Varieties and Parameter Spaces for Gorenstein Ideals, The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995), pp. 2–114 (1996)Google Scholar
  14. 14.
    Harbourne, B.: Points in Good Position in \({\mathbb{P}^2}\), Zero-dimensional schemes (Ravello, 1992), pp. 213–229 (1994)Google Scholar
  15. 15.
    Harbourne B., Schenck H., Seceleanu A.: Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property. J. Lond. Math. Soc. (2) 84(3), 712–730 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  17. 17.
    Iarrobino A.: Compressed algebras: Artin algebras having given socle degrees and maximal length. Trans. Am. Math. Soc. 285(1), 337–378 (1984)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Iarrobino A.: Inverse system of a symbolic power. III. Thin algebras and fat points. Compositio Math. 108(3), 319–356 (1997)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kolesnikov, A., Sorokina, T.: Multivariate C 1-continuous splines on the Alfeld split of a simplex., to appear in Approximation Theory XIV: San Antonio (2013)
  20. 20.
    Laface A., Ugaglia L.: Standard classes on the blow-up of \({\mathbb{P}^n}\) at points in very general position. Comm. Algebra 40(6), 2115–2129 (2012)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Lai, M-J., Schumaker, L.L.: Spline Functions on Triangulations, Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007)Google Scholar
  22. 22.
    Lau W.: A lower bound for the dimension of trivariate spline spaces. Constr. Approx. 23(1), 23–31 (2006)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Migliore J., Miró-Roig R., Nagel U.: On the weak Lefschetz property for powers of linear forms. Algebra Number Theory 6(3), 487–526 (2012)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Mourrain B., Villamizar N.: Homological techniques for the analysis of the dimension of triangular spline spaces. J. Symb. Comput. 50, 564–577 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Nagata, M.: On rational surfaces. II. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33:271–293 (1960/1961)Google Scholar
  26. 26.
    Schenck H.: A spectral sequence for splines. Adv. Appl. Math. 19(2), 183–199 (1997)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Schenck H., Stillman M.: A family of ideals of minimal regularity and the Hilbert series of \({C^r(\hat{\Delta})}\). Adv. Appl. Math. 19(2), 169–182 (1997)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Schenck, H., Stillman, M.: Local cohomology of bivariate splines. J. Pure Appl. Algebra 117/118:535-548. Algorithms for algebra (Eindhoven, 1996) (1997)Google Scholar
  29. 29.
    Shan, J.: Dimension of C 2 trivariate splines on cells., to appear in Approximation Theory XIV: San Antonio (2013)
  30. 30.
    Ženíšek A.: Polynomial approximation on tetrahedrons in the finite element method. J. Approx. Theory 7, 334–351 (1973)CrossRefMATHGoogle Scholar
  31. 31.
    Zlámal M.: On the finite element method. Numer. Math. 12, 394–409 (1968)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Inria Sophia Antipolis MéditerranéeSophia AntipolisFrance
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

Personalised recommendations