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Zeros, multiplicities, and idempotents for zero-dimensional systems

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Algorithms in Algebraic Geometry and Applications

Part of the book series: Progress in Mathematics ((PM,volume 143))

Abstract

We want to propose alternative computational methods for dealing with the following three classical problems in the study of zero-dimensional systems, rephrased in the context of finite-dimensional algebras over a field k of characteristic zero. It is the main feature of our approach to adapt to the affine case the concept of the u-Chow form (or u-resultant) which was developed in the projective case (and has been used by several authors, e.g., [Ca] and [Re]).

Partially supported by POSSO, Esprit BRA 6846, 2nd and 4th author also acknowledge support from DFG

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References

  1. S. Basu, R. Pollack, M.-F. Roy: A new algorithm to find a point in every cell defined by a family of polynomials, quantifier elimination and cylindrical algebraic decomposition, B. Caviness, J. Johnson (Eds.), Springer- Verlag, to appear.

    Google Scholar 

  2. E. Becker, J.-P. Cardinal, M.-F. Roy, Z. Szafraniec: Multivariate be- zoutians, Kronecker symbol and Eisenbud-Levine formula. In this volume.

    Google Scholar 

  3. E. Becker, T. Wörmann: On the trace formula for quadratic forms, recent advances in real algebraic geometry and quadratic forms, Proceedings of the RAGSQUAD year, Berkeley 1990–1991, W. B. Jacob, T.-Y. Lam, R. O. Robson (editors), Contemporary Mathematics 155, pp. 271–291 (1994).

    Google Scholar 

  4. E. Becker, T. Wörmann: Radical computations of zero-dimensional ideals and real root counting, to appear in Mathematics and Computers in Simulation.

    Google Scholar 

  5. J.F. Canny: Some algebraic and geometric computations in PSPACE, In Proc. Twentieth ACM Symp. on Theory of Computing, pp. 460 -467 (1988).

    Google Scholar 

  6. H. Caprasse, Departement d’Astronomie et d’Astrophysique, Institut de Physique B5, Sart Tilman, B-4000 LIEGE.

    Google Scholar 

  7. A. Chistov, D. Grigor’ev: Subexponential time solving of systems of algebraic equations, Preprint LOMIE-9–83, Leningrad.

    Google Scholar 

  8. J.C. Faugére, P. Gianni, D. Lazard, T. Mora: Efficient computation of zero-dimensional Gröbner-Bases by change of orderings. unpublished manuscript (1989).

    Google Scholar 

  9. A. Lecki, Z. Szafraniec: Application of the Eisenbud-Levine’s theorem to real algebraic geometry, Proceedings of MEGA 1992.

    Google Scholar 

  10. R. Narasimhan: Introduction to the theory of analytic spaces, Lecture Notes in Mathematics 25, Springer (1966).

    MATH  Google Scholar 

  11. P. Pedersen, M.-F. Roy, A. Szpirglas: Counting real zeros in the multivariate case, Computational Algebraic Geometry, Frédéric Eyssette, André Galligo (editors), pp. 203–223 (1993), Birkhäuser.

    Google Scholar 

  12. J. Renegar: On the computational complexity and geometry of the first- order theory of the reals, parts I, II and III. Journal of Symbolic Computation, 13(3), pp. 255–352, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. M. Ruiz: The basic theory of power series, Advanced Lectures in Mathematics, Vieweg, Braunschweig/Wiesbaden (1993).

    Google Scholar 

  14. G. Scheja, U. Storch: Lehrbuch der Algebra, Band 2 (1988). Teubner.

    Google Scholar 

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© 1996 Birkhäuser Verlag Basel/Switzerland

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Alonso, ME., Becker, E., Roy, M.F., Wörmann, T. (1996). Zeros, multiplicities, and idempotents for zero-dimensional systems. In: González-Vega, L., Recio, T. (eds) Algorithms in Algebraic Geometry and Applications. Progress in Mathematics, vol 143. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9104-2_1

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  • DOI: https://doi.org/10.1007/978-3-0348-9104-2_1

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9908-6

  • Online ISBN: 978-3-0348-9104-2

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