Skip to main content
SpringerLink
Log in

Generalized principal lattices and cubic pencils

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bix, R.: Conics and Cubics: A Concrete Introduction to Algebraic Curves. Springer, New York (1998)

    MATH  Google Scholar 

  2. Busch, J.R.: A note on Lagrange interpolation in \(\bf R^2\). Rev. Unión Mat. Argent. 36, 33–38 (1990)

    MATH  MathSciNet  Google Scholar 

  3. Carnicer, J.M., García-Esnaola, M.: Lagrange interpolation on conics and cubics. Comput. Aided Geom. Des. 19, 313–326 (2002)

    Article  MATH  Google Scholar 

  4. Carnicer, J.M., Gasca, M.: Planar configurations with simple lagrange formulae. In: Lyche, T., Schumaker, L.L. (eds.), Mathematical Methods in CAGD: Oslo 2000, Vanderbilt University Press, Nashville, pp. 55–62 (2001)

    Google Scholar 

  5. Carnicer, J.M., Gasca, M.: A conjecture on multivariate polynomial interpolation. Rev. R. Acad. Cien. Serie A. Mat. 95, 145–153 (2001)

    MATH  MathSciNet  Google Scholar 

  6. Carnicer, J.M., Gasca, M.: On Chung and Yao’s geometric characterization for bivariate polynomial interpolation. In: Lyche, T., Mazure, M.-L., Schumaker, L.L. (eds.), Curve and Surface Design: St. Malo 2002, pp. 21–30. Nashboro Press, Nashville (2003)

    Google Scholar 

  7. Carnicer, J.M., Gasca, M.: Generation of lattices of points for bivariate interpolation. Numer. Algorithms 39, 69–79 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carnicer, J.M., Gasca, M.: Interpolation on lattices generated by cubic pencils. Adv. Comp. Math. 24, 113–130 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carnicer, J.M., Gasca, M., Sauer, T.: Interpolation lattices in several variables. Numer. Math. 102, 559–581 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carnicer, J.M., Godés, C.: Geometric characterization and generalized principal lattices. J. Approx. Theory 143, 2–14 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolation. SIAM J. Numer. Anal. 14, 735–743 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  12. Eisenbud, D., Green, M., Harris, J.: Cayley–Bacharach theorems and conjectures. Bull. Am. Math. Soc. 33, 295–323 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gasca, M., Maeztu, J.I.: On Lagrange and Hermite interpolation in \({\bf R}^n\). Numer. Math. 39, 1–14 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lee, S.L., Phillips, G.M.: Construction of lattices for lagrange interpolation in projective space. Constr. Approx. 7, 283–297 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nicolaides, R.A.: On a class of finite elements generated by Lagrange interpolation. SIAM J. Numer. Anal. 9, 435–445 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  16. Walker, R.J.: Algebraic Curves. Springer, New York (1978)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Zaragoza, Pedro Cerbuna 12, Zaragoza, 50009, Spain

    J. M. Carnicer

  2. University of Zaragoza, Escuela Politécnica Superior, Carretera de Cuarte s/n, Huesca, 22071, Spain

    C. Godés

Authors
  1. J. M. Carnicer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Godés
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. Carnicer.

Additional information

Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carnicer, J.M., Godés, C. Generalized principal lattices and cubic pencils. Numer Algor 44, 133–145 (2007). https://doi.org/10.1007/s11075-007-9091-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-007-9091-5

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation