Numerische Mathematik

, Volume 70, Issue 3, pp 311–329 | Cite as

Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems

  • H. Michael Möller
  • Hans J. Stetter


The eigenproblem method calculates the solutions of systems of polynomial equations \( f_1(x_1, \ldots , x_s)=0,\ldots,f_m(x_1, \ldots , x_s)=0\). It consists in fixing a suitable polynomial \( f \) and in considering the matrix \( A_f \) corresponding to the mapping \( [p] \mapsto [f\cdot p] \) where the equivalence classes are modulo the ideal generated by \( f_1, \ldots , f_m.\) The eigenspaces contain vectors, from which all solutions of the system can be read off. This access was investigated in [1] and [16] mainly for the case that \(\) is nonderogatory. In the present paper, we study the case where \( f_1, \ldots , f_m \) have multiple zeros in common. We establish a kind of Jordan decomposition of \( A_f \) reflecting the multiplicity structure, and describe the conditions under which \( A_f \) is nonderogatory. The algorithmic analysis of the eigenproblem in the general case is indicated.

Mathematics Subject Classification (1991): 12D10, 26D10, 30C15, 65H10, 65H15 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • H. Michael Möller
    • 1
  • Hans J. Stetter
    • 2
  1. 1.FB Mathematik der FernUniversität, D-58084 HagenDE
  2. 2.Institut für Numerische Mathematik, Technische Universität, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria AT

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