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Multiplication Matrices and Ideals of Projective Dimension Zero

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Abstract

We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

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References

  1. Abbott J., Bigatti A., Kreuzer M., Robbiano L.: Computing ideals of points. J. Symb. Comput. 30(4), 341–356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abbott J., Kreuzer M., Robbiano L.: Computing Zero-Dimensional Schemes. J. Symb. Comp. 39(1), 31–49 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alonso, M.E., Mora, T., Niesi, G., Raimondo, M.: An algorithm for computing analytic branches of space curves at singular points. In: Proceedings 1992 International Workshop on Mathematics Mechanization. International Academic Publishers, Pekin, pp. 135–166 (1992)

  4. Auzinger, W., Stetter, H.J.: An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. I.S.N.M. 86, 11–30, Birkhäuser (1988)

    Google Scholar 

  5. Atiyah M.F., Macdonald I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co, Reading (1969)

    MATH  Google Scholar 

  6. Björck G., Fröberg F.: A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots. J. Symb. Comput. 12(3), 329–336 (1991)

    Article  MATH  Google Scholar 

  7. Buchberger B., Möller M.: The construction of multivariate polynomials with preassigned zeroes. Computer Algebra (Marseille, 1982), Lecture Notes in Comput. Sci. 144, 24–31 (1982)

    Article  Google Scholar 

  8. Coppersmith D., Winograd S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Corless R.M.: Editor’s corner: Gröbner bases and matrix eigenproblems. SIGSAM Bull. 30(4), 26–32 (1996)

    Article  MathSciNet  Google Scholar 

  10. Corless R.M., Gatermann K., Kotsireas I.S.: Using symmetries in the eigenvalue method for polynomial systems. J. Symb. Comput. 44(11), 1536–1550 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Corless, R.M., Gianni, P.M., Trager, B.M.: A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 133–140 (1997)

  12. Faugère J.C., Gianni P., Lazard D., Mora T.: Efficient computation of zero-dimensional Gröbner basis by change of ordering. J. Symb. Comp. 16(4), 329–344 (1993)

    Article  MATH  Google Scholar 

  13. Geramita A.V., Maroscia P., Roberts L.: The Hilbert function of a reduced k-algebra. J. London Math. Soc. (2) 28(3), 443–452 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gotzmann G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z. 158(1), 61–70 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gröbner W.: Algebraische Geometrie. Bibliographisches Institut Mannheim, Mannheim (1970)

    MATH  Google Scholar 

  16. Kreuzer M.: Computing Hilbert-Kunz functions of 1-dimensional graded rings. Universitatis Iagellonicae Acta Mathematica 45, 81–95 (2007)

    MathSciNet  Google Scholar 

  17. Kreuzer M., Robbiano L.: Computational Commutative Algebra 2. Springer, Heidelberg (2005)

    Google Scholar 

  18. Lazard D.: Algèbre linéaire sur K[X 1, . . . ,X n ] et élimination. Bull. Soc. Math. France 105, 165–190 (1977)

    MathSciNet  MATH  Google Scholar 

  19. Lazard D.: Resolution des systemes d’equations algebriques. Theoret. Comput. Sci. 15, 77–110 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lundqvist, S.: An algorithm to determine the Hilbert series for graded associative algebras. Research Reports in Mathematics, vol. 3, pp. 1–19. Stockholm University (2005)

  21. Lundqvist S.: Complexity of comparing monomials and two improvements of the Buchberger–Möller algorithm. MMISC, Lecture Notes in Comput. Sci. 5393, 105–125 (2008)

    Article  MathSciNet  Google Scholar 

  22. Lundqvist S.: Vector space bases associated to vanishing ideals of points. J. Pure Appl. Alg. 214(4), 309–321 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lundqvist S.: Non-vanishing forms in projective space over finite fields. J. Comm. Alg. 2(4), 435–443 (2010)

    MathSciNet  Google Scholar 

  24. Marinari M.G., Möller H.M., Mora T.: Gröbner bases of ideals defined by functionals with an application to ideals of projective points. Appl. Algebra Eng. Commun. Comput. 4(2), 103–145 (1993)

    Article  MATH  Google Scholar 

  25. Möller H.M., Stetter H.: Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems. Numer. Math. 70(3), 311–329 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Möller H.M., Tenberg R.: Multivariate polynomial system solving using intersection of eigenspaces. J. Symb. Comput. 32(5), 513–531 (2001)

    Article  MATH  Google Scholar 

  27. Mourrain B.: Isolated points, duality and residues. J. Pure Appl. Alg. 117, 118, 469–493 (1997)

    Article  MathSciNet  Google Scholar 

  28. Robbiano L.: Introduction to the theory of Hilbert functions. Queen’s Papers in Pure and Appl. Math. 85, B1–B26 (1990)

    Google Scholar 

  29. Stetter H.: Numerical Polynomial Algebra, xv+472 pp. SIAM, Philadelphia (2004)

    Book  Google Scholar 

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  1. Department of Mathematics, Stockholm University, 106 91, Stockholm, Sweden

    Samuel Lundqvist

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  1. Samuel Lundqvist
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Lundqvist, S. Multiplication Matrices and Ideals of Projective Dimension Zero. Math.Comput.Sci. 6, 43–59 (2012). https://doi.org/10.1007/s11786-012-0108-7

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  • DOI: https://doi.org/10.1007/s11786-012-0108-7

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