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Algebra pp 377-412 | Cite as

Algebraic Spaces

  • Serge Lang
Part of the Graduate Texts in Mathematics book series (GTM, volume 211)

Abstract

This chapter gives the basic results concerning solutions of polynomial equations in several variables over a field k. First it will be proved that if such equations have a common zero in some field, then they have a common zero in the algebraic closure of k, and such a zero can be obtained by the process known as specialization. However, it is useful to deal with transcendental extensions of k as well. Indeed, if p is a prime ideal in k[X] = k[X 1, …, X n ], then k[X]/p is a finitely generated ring over k, and the images x i of X t in this ring may be transcendental over k, so we are led to consider such rings.

Keywords

Projective Space Generic Point Prime Ideal Commutative Ring Polynomial Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [BeY.
    91]_C. Berenstein and A. Yger, Effective Bezout identities in Q[z 1,..., z n], Acta Math. 166 (1991), pp. 69–120MathSciNetMATHCrossRefGoogle Scholar
  2. [Br 87]
    D. Brownawell, Bounds for the degree in Nullstellensatz, Ann. of Math. 126 (1987), pp. 577–592MathSciNetMATHCrossRefGoogle Scholar
  3. [Br 88]
    D. Brownawell, Local diophantine nullstellen inequalities, J. Amer. Math. Soc. 1 (1988), pp. 311–322MathSciNetMATHCrossRefGoogle Scholar
  4. [Br 89]
    D. Brownawell, Applications of Cayley-Chow forms, Springer Lecture Notes 1380: Number Theory, Ulm 1987, H. P. Schlickewei and E. Wirsing (eds.), pp. 1-18Google Scholar
  5. [Ko 88]
    J. Kollar, Sharp effective nullstellensatz, J. Amer. Math. Soc. 1No. 4 (1988), pp. 963–975MathSciNetMATHCrossRefGoogle Scholar

Bibliography

  1. [Jo 80]
    J. P. Jouanolou, Idéaux résultants, Advances in Mathematics 37No. 3 (1980), pp. 212–238MathSciNetMATHCrossRefGoogle Scholar
  2. [Jo 90]
    J. P. Jouanolou, Le formalisme du résultant, Advances in Mathematics 90No. 2 (1991) pp. 117–263MathSciNetMATHCrossRefGoogle Scholar
  3. [Jo 91]
    J. P. Jouanolou, Aspects invariants de l’élimination, Départment de Mathématiques, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  4. [Ma 16]
    F. Macaulay, The algebraic theory of modular systems, Cambridge University Press, 1916Google Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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