Duality methods for the membership problem

  • Alicia Dickenstein
  • Carmen Sessa
Part of the Progress in Mathematics book series (PM, volume 94)


The classical problem of deciding membership to arbitrary polynomial ideals is EXPSPACE complete. Moreover, the problem of finding a representation of a polynomial by generators of a given ideal may involve doubly exponential (in the number of variables) degrees ([16]). The same difficulty arises when computing Gröebner bases of arbitrary polynomial ideals ([11]). This means that all known techniques to decide membership and to find representations of polynomials with respect to a given ideal lead to doubly exponential (sequential time) worst case complexities. However, if the geometry of the underlying algebraic variety is particularly simple, e.g. if the given ideal is zero dimensional or complete intersection, algorithms of considerably lower complexity can be found (see e.g. [7], [9]). The improvements are due to recent progress concerning affine versions of the effective Nullstellensatz (compare [18] and the references given there).


Complete Intersection Polynomial Ideal Regular Sequence Duality Method Membership Problem 
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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Alicia Dickenstein
    • 1
  • Carmen Sessa
    • 1
  1. 1.Departamento de Matemática — FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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