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Variants of the Effective Nullstellensatz and Residue Calculus

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Notions of Positivity and the Geometry of Polynomials

Part of the book series: Trends in Mathematics ((TM))

Abstract

We describe how one can obtain effective versions of the Nullstellensatz and variations by a combination of residue calculus and a geometric estimate for so-called distinguished varieties.

Mathematics Subject Classification (2000). 14Q20, 32A27, 32B99.

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References

  1. M. Andersson: Residue currents and ideals of holomorphic functions, Bull. Sci. Math. 128, (2004), 481–512.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Andersson: The membership problem for polynomial ideals in terms of residue currents, Ann. Inst. Fourier 56 (2006), 101–119.

    MATH  Google Scholar 

  3. M. Andersson: Explicit versions of the Briancon-Skoda theorem with variations, Michigan Math. J. 54 ( 2 )(2006), 361–373.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Andersson & E. Götmark: Explicit representation of membership in polynomial ideals, Math. Ann. 349 (2011), 345–365.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Andersson & E. Wulcan: Decomposition of residue currents, J. Reine Angew. Math. 638 (2010), 103–118.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Andersson & E. Wulcan: On the effective membership problem on singular varieties, Preprint, Göteborg 2011.

    Google Scholar 

  7. J. Briançon & H. Skoda: Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de ℂn, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949–951.

    MATH  Google Scholar 

  8. J-P Demailly: Complex Analytic and Differential Geometry, Monograph Grenoble (1997).

    Google Scholar 

  9. L. Ein & R. Lazarsfeld: A geometric effective Nullstellensatz, Invent. math. 135 (1999), 427–448.

    Article  MathSciNet  Google Scholar 

  10. W. Fulton: Intersection theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1984. xi+470 pp.

    Google Scholar 

  11. M. Hickel: Solution d’une conjecture de C. Berenstein-A. Yger et invariants de contact à l’infini, Ann. Inst. Fourier 51 (2001), 707–744.

    MathSciNet  MATH  Google Scholar 

  12. Z. Jelonek: On the effective Nullstellensatz, Invent. math. 162 1–17 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Kollár: Sharp effective Nullstellensatz, J. American Math. Soc. 1 (1988), 963–975.

    Article  MATH  Google Scholar 

  14. R. Lazarsfeld: Positivity in algebraic geometry I and II, Springer-Verlag 2004.

    Google Scholar 

  15. F.S. Macaulay: The algebraic theory of modular systems, Cambridge Univ. Press, Cambridge 1916.

    MATH  Google Scholar 

  16. M. Nöther: Über einen Satz aus der Theorie der algebraischen Functionen, Math. Ann. (1873), 351–359.

    Google Scholar 

  17. M. Sombra: A sparse effective Nullstellensatz, Adv. in Appl. Math. 22 (1999) 271–295.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Passare & A. Tsikh & A. Yger: Residue currents of the Bochner-Martinelli type, Publ. Mat. 44 (2000), 85–117.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, S-412 96, Gothenburg, Sweden

    Mats Andersson & Elizabeth Wulcan

Authors
  1. Mats Andersson
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  2. Elizabeth Wulcan
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Corresponding author

Correspondence to Mats Andersson .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    Petter Brändén

  2. , Mathematical Department, Stockholm University, Stockholm, SE-106 91, Sweden

    Mikael Passare

  3. , Department of Mathematics, University of California, Santa Barbara, 93106-3080, California, USA

    Mihai Putinar

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Andersson, M., Wulcan, E. (2011). Variants of the Effective Nullstellensatz and Residue Calculus. In: Brändén, P., Passare, M., Putinar, M. (eds) Notions of Positivity and the Geometry of Polynomials. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0348-0142-3_2

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