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Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 101–126 | Cite as

From spline approximation to Roth’s equation and Schur functors

  • Ján Mináč
  • Stefan O. TohǎneanuEmail author
Article

Abstract

Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ, for d ≥ 3r + 1. Schenck and Stiller conjectured that this formula actually holds for all d ≥ 2r + 1. Up to this moment there was not known a single example where one could show that the bound d ≥ 2r + 1 is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld–Schumaker formula does not hold if d = 2r), except that the proof that this formula actually works if d ≥ 2r + 1 has been a challenge until now when we finally show it to be true. The interesting subtle connections with representation theory, matrix theory and commutative and homological algebra seem to explain why this example presented such a challenge. Thus in this paper we present the first example when it is known that the bound d ≥ 2r + 1 is sharp for asserting the validity of the Alfeld–Schumaker formula.

Mathematics Subject Classification (2000)

Primary 41A15 Secondary 13D40 52B20 15A23 

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References

  1. 1.
    Alfeld P., Schumaker L.: On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1. Numer. Math. 57, 651–661 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bhatia, R.: Positive Definite Matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)Google Scholar
  3. 3.
    Billera, L., Rose, L.: Gröbner basis methods for multivariate splines. In: Mathematical Methods in Computer Aided Geometric Design, pp. 93–104. Academic Press, Boston (1989)Google Scholar
  4. 4.
    Billera L., Rose L.: A dimension series for multivariate splines. Discret. Comput. Geom. 6, 107–128 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Berlin (1998)Google Scholar
  6. 6.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995)Google Scholar
  7. 7.
    Fulton, W., Harris, J.: Representation Theory. Springer, New York (1991)Google Scholar
  8. 8.
    Gantmacher, F.R.: The Theory of Matrices, vol. 1. Chelsea Publishing Company, New York (1977)Google Scholar
  9. 9.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Clarendon Press, Oxford (1995)Google Scholar
  10. 10.
    Nicholson, W.K.: Elementary Linear Algebra with Applications, 2nd edn. PWS-KENT Publishing Company, Boston (1986)Google Scholar
  11. 11.
    Pinkus, A.: Totally Positive Matrices. Cambridge University Press, Cambridge (2010)Google Scholar
  12. 12.
    Prasolov, V.: Problems and Theorems in Linear Algebra. Translations of Mathematical Monographs, vol. 134. AMS, Providence (1994)Google Scholar
  13. 13.
    Roth W.E.: The equations AXYB = C and AXXBC in matrices. Proc. Am. Math. Soc. 3, 392–396 (1952)zbMATHGoogle Scholar
  14. 14.
    Schenck H., Stiller P.: Cohomology vanishing and a problem in approximation theory. Manuscr. Math. 107, 43–58 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Schenck H., Stillman M.: A Family of ideals of minimal regularity and the Hilbert series of \({\mathcal{C}^r(\hat{\Delta})}\) . Adv. Appl. Math. 19, 169–182 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Schenck H., Stillman M.: Local cohomology of bivariate splines. J. Pure Appl. Algebra 117(118), 535–548 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shi X., Wang T., Yin B.: Splines on generalized quasi-cross-cut partitions. J. Comput. Appl. Math. 96, 139–147 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Stanley R.: Hilbert function of graded algebras. Adv. Math. 28, 57–83 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Tohaneanu S.: Smooth planar r-splines of degree 2r. J. Approx. Theory 132, 72–76 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsWestern UniversityLondonCanada
  2. 2.Department of MathematicsWestern UniversityLondonCanada

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