# Duality methods for the membership problem

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

## Abstract

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).

## Keywords

Complete Intersection Polynomial Ideal Regular Sequence Duality Method Membership Problem
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## References

1. [1]
Angeniol, B., Résidus et Effectivité, Preprint (1983).Google Scholar
2. [2]
Berenstein, C., Yger, A., Bounds for the degrees in the division problem, Preprint Univ. of Maryland (1989).Google Scholar
3. [3]
Brownawell, W.D., Bounds for the degrees in the Nullstellensatz, Annals of Math. 126 (1987), 577–591.
4. [4]
Caniglia, L., Galligo, A., Heintz, J., Some new effectivity bounds in Computational Geometry, in “Applied Algebra, Algebraic Algorithms and Error Correcting Codes. Proc. 6th Int’l Conf., Rome 1988,” (Ed. T. Mora), Springer LN Comput. Sci. 357, 1989, pp. 131–151.Google Scholar
5. [5]
Chistov, A.L., Grigor’ev, D.Yu., Subexponential time solving systems of algebraic equations, LOMI preprints E-9-83 E-10-83, Leningrad (1983).Google Scholar
6. [6]
Coleff, N., Herrera, M., “Les Courants Residueis Associés à une Forme Meromorphe,” Springer LN Math. 633, 1978.Google Scholar
7. [7]
Dickenstein, A., Fitchas, N., Giusti, M., Sessa, C., The membership problem for unmixed polynomial ideals is solvable in single exponential time, in “Discrete Applied Algebra, Proc. AAECC-7, Toulouse 1989” (to appear).Google Scholar
8. [8]
Dickenstein, A., Sessa, C., Canonical Representatives in Moderate Cohomology, Invent. Math. 80 (1985), 417–434.
9. [9]
Dickenstein, A., Sessa, C., An Effective Residual Criterion for the Membership Problem in C[z 1,…, z n], J. Pure Appl. Algebra (to appear).Google Scholar
10. [10]
von zur Gathen, J., Parallel arithmetic computations. A survey, in “Proc. 13th Symp. MFCS 1986,” Springer LN Comput. Sci. 233, 1986, pp. 93–112.Google Scholar
11. [11]
Giusti, M., Complexity of standard bases in projective dimension zero, Preprint Ecole Polytechnique Paris (1987).Google Scholar
12. [12]
Griffiths, P., Harris, J., “Principles of Algebraic Geometry,” John Wiley & Sons, 1978.Google Scholar
13. [13]
Hartshorne, R., “Residues and Duality,” Springer L.N. Math. 20, 1966.Google Scholar
14. [14]
Lazard, D., Algèbre linéaire sur K[x 1,…, x n]_et élimination, Bull. Soc. Math. France 105 (1977), 165–190.
15. [15]
Lipman, J., Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117 (1984).Google Scholar
16. [16]
Mayr, E., Meyer, A., The complexity of the word problem for commutative semigroups and polynomial ideals, Advances in Math. 46 (1982), 305–329.
17. [17]
Serre, J.P., Géometrie Algébrique et Géometrie Analytique, (G.A.G.A.), Annales de l’Institut Fourier VI (1956), 1–42.Google Scholar
18. [18]
Teissier, B., Résultats récents d’algèbre commutative effective, in “Séminaire Bourbaki, 42ème année, 1989-90,” n° 718, pp. 1–19.Google Scholar
19. [19]
Zariski, O., Samuel, P., “Commutative Algebra,” Vol. 1, Van Nostrand, New York, 1958.