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Gröbner bases of ideals defined by functionals with an application to ideals of projective points

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Abstract

In this paper we study 0-dimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morphisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, generalizing the Buchberger-Möller algorithm for computing a basis of an ideal vanishing at a set of points and the FGLM basis conversion algorithm.

As an application to Algebraic Geometry, we show how to compute in polynomial time a minimal basis of an ideal of projective points.

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References

  1. Backelin, J.: Communication at MEGA '90

  2. Backelin, J., Fröberg, R.: How we proved that there exactly 924 cyclic 7-roots, Proc. ISSAC 91, 103–111, ACM (1991)

    Google Scholar 

  3. Becker, T., Weispfenning, V.: The Chinese remainder, multivariate interpolation and Gröbner bases. Proc. ISSAC 91, 64–69, ACM (1991)

    Google Scholar 

  4. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Innsbruck, (1965)

  5. Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory in Recent trends in multidimensional systems theory, Bose, N. K. (ed.) Dordrecht: Reidel 1985

    Google Scholar 

  6. Buchberger, B.: A criterion for detecting unnecessary reductions in the construction of Gröbner bases. Proc. Eurosam 79. Lecture Notes in Computer Science vol. 72 pp. 3–21. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  7. Buchberger, B., Möller, H. M.: The construction of multivariate polynomials with preassigned zeroes, Proc. EUROCAM 82, Lecture Notes in Computer Science vol. 144 pp. 24–31. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  8. Cerlienco, L., Mureddu, M.: Combinatorial algorithms for polynomial interpolation in dimension ≧ 2 (in preparation)

  9. Eliahou, S.: Minimal syzygies of monomial ideals and Gröbner bases (preprint)

  10. Faugere, J. C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symb. Comp. (submitted) (1989)

  11. Gebauer, R., Möller, H. M.: On an installation of Buchberger's Algorithm, J. Symb. Comp.6, 275–286 (1988)

    Google Scholar 

  12. Gianni, P., Mora, T.: Algebraic solution of systems of polynomial equations using Gröbner bases. Lecture Notes in Computer Science vol. 356, pp. 247–257. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  13. Giusti, M.: Complexity of standard bases in projective dimension zero II. Proc. AAECC 8. Lecture Notes in Computer Science vol. 508, pp. 322–328. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  14. Giusti, M., Heintz, J.: Algorithmes — disons rapides — pour la décomposition d'une variété en composantes irréductibles et équidimensionnelles, Proc. MEGA '90. Basel: Birkhäuser 1991

  15. Gröbner, W.: Algebraische Geometrie, Vol. 2, Bibliographisches Institut Mannheim (1967–1968)

  16. Heuser, H. G.: Functional analysis. New York: Wiley 1982

    Google Scholar 

  17. Lakshman, Y. N.: A single exponential bound on the complexity of computing Gröbner bases of zero dimensional ideals. Proc. MEGA '90. Basel: Birkhauser 1991

    Google Scholar 

  18. Lakshman, Y. N.: On the complexity of computing Gröbner bases of zero dimensional ideals, Ph.D. Thesis, Rensselaer Polytechnique Institute, Troy (1990)

    Google Scholar 

  19. Lazard, D.: Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, Proc. Eurocal 83. Lecture Notes in Computer Science vol. 162, pp. 146–156. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  20. Marihari, M. G., Möller, H. M., Mora, T.: Gröbner bases of ideals given by dual bases. Proc. ISSAC 91, ACM 55–63 (1991)

    Google Scholar 

  21. Möller, H. M., Mora, T.: New constructive methods in classical ideal theory. J. Algebra100, 138–178 (1986)

    Google Scholar 

  22. Ramella, L: Algoritmi di computer algebra relativi agli ideali di punti dello spazio proiettivo, Ph.D. Thesis, Univ. Napoli (1990)

  23. Robbiano, L.: Bounds for degrees and number of elements in Gröbner bases. Proc. AAECC 8. Lecture Notes in Computer Science vol. 508. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

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Dedicated to our friend Mario Raimondo

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Marinari, M.G., Möller, H.M. & Mora, T. Gröbner bases of ideals defined by functionals with an application to ideals of projective points. AAECC 4, 103–145 (1993). https://doi.org/10.1007/BF01386834

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