Local Decomposition Algorithms

@inproceedings{Alonso1990LocalDA,
  title={Local Decomposition Algorithms},
  author={Maria Emilia Alonso and Teo Mora and Mario Di Raimondo},
  booktitle={AAECC},
  year={1990}
}
INTRODUCTION For an ideal I ⊂ P := k[X 1 ,…,X n ], many algorithms using Gröbner techniques are a direct consequence of the definition itself of Gröbner basis: among them we can list algorithms for computing syzygies, dimension, minimal bases, free resolutions, Hilbert function and other algebro-geometric invariants of an ideal; the explicit knowledge of syzygies allows moreover to compute ideal intersection and quotients. Other algorithms require a specific property of Gröbner bases w.r.t. of… 

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References

SHOWING 1-10 OF 11 REFERENCES

Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals

TLDR
Borders are given for the complexity of the above problems which are simply exponential in the number n of variables in the one-dimensional case, and it is shown that in the general case the first two problems are doubly exponential only in the dimension of the ideal.

Computing with algebraic series

TLDR
An effective version of Weierstrass theorems are given, which allow us to have effective elimination theory for algebraic series and an effective Noether Normalization Lemma for Noether normal position of an ideal of algebraic formal power series.

Solving Zero-Dimensional Algebraic Systems

  • D. Lazard
  • Computer Science, Mathematics
    J. Symb. Comput.
  • 1992

Algorithmes – disons rapides – pour la décomposition d’une variété algébrique en composantes irréductibles et équidimensionnelles

Resume. Nous decrivons dans cet article deux algorithmes qui construisent les decompositions irreductible et equidimensionnelle d’une variete algebrique affine (ou projective), definie par un

An introduction to the tangent cone algorithm, Issues in nonlinear geometry and robotics

  • An introduction to the tangent cone algorithm, Issues in nonlinear geometry and robotics

conferences at COCOA I (1986), Luminy (1988) and elsewhere A Jacobian method for finding the radical of an ideal

  • conferences at COCOA I (1986), Luminy (1988) and elsewhere A Jacobian method for finding the radical of an ideal
  • 1989