The construction of multivariate polynomials with preassigned zeros

  • H. M. Möller
  • B. Buchberger
1. Algorithms I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 144)


We present an algorithm for constructing a basis of the ideal of all polynomials, which vanish at a preassigned set of points {y1,...,ym} ⊂ Kn, K a field. The algorithm yields also Newton-type polynomials for pointwise interpolation. These polynomials admit an immediate construction of interpolating polynomials and allow to shorten the algorithm, if it is applied to an enlarged set {y1,...,ym1} ⊂ Kn, m1>m.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Birkhoff, The algebra of multivariate interpolation, in: Constructive Approaches to Mathematical Models (ed.: C.V. Coffman and G.J. Fix), Academic Press 1979, p. 345–363.Google Scholar
  2. [2]
    B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM Bulletin 39, August 1976, p. 19–29.Google Scholar
  3. [3]
    Ph. Defert and J.P. Thiran, Chebyshev approximation by multivariate polynomials, Report 80/10, Facultés Universitaires de Namur, Belgium, 1980.Google Scholar
  4. [4]
    C. Günther, IPOL — Ein Fortran-Programm zur zweidimensionalen Interpolation, Report KFK 2175, Kernforschungszentrum Karlsruhe, W.-Germany, November 1975.Google Scholar
  5. [5]
    H.M. Möller, Mehrdimensionale Hermite-Interpolation und numerische Integration, Math. Z. 148 (1976), 107–118.CrossRefGoogle Scholar
  6. [6]
    H.J. Schmid, Interpolatory cubature formulae and real ideals, in: Quantitative Approximation (ed.: Ronald A. deVore and K. Scherer), Academic Press 1980, p. 245–254.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • H. M. Möller
    • 1
  • B. Buchberger
    • 2
  1. 1.Fernuniversität HagenHagenW.-Germany
  2. 2.Universität LinzLinzAustria

Personalised recommendations