Skip to main content
SpringerLink
Log in

Sparse effective membership problems via residue currents

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max Nöther’s AF + BG Theorem, Macaulay’s Theorem, and Kollár’s Effective Nullstellensatz, as well as recent results by Hickel and Andersson–Götmark.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahlberg, D.: Some variants of the Max Nöther and Macaulay Theorems. Master’s Thesis. Chalmers University of Technology and Göteborg University (2006)

  2. Andersson M.: Residue currents and ideals of holomorphic functions. Bull. Sci. Math. 128(6), 481–512 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Andersson, M., Götmark, E.: Explicit representation of membership of polynomial ideals. Math. Ann. (to appear)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Berenstein, C.A., Gay, R., Vidras, A., Yger, A.: Residue currents and Bezout identities. Progress in Mathematics, vol. 114. Birkhäuser Verlag, Basel (1993)

  7. Berenstein C.A., Vidras A., Yger A.: Analytic residues along algebraic cycles. J. Complexity 21(1), 5–42 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Björk, J.-E.: Residues and \({\mathcal D}\)-modules. In: The legacy of Niels Henrik Abel, pp. 605–651. Springer, Berlin (2004)

  9. Briançon J., Skoda H.: Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de \({\mathbb C^n}\). C. R. Acad. Sci. Paris Sér. A 278, 949–951 (1974)

    MATH  Google Scholar 

  10. Brownawell W.D.: Bounds for the degrees in the Nullstellensatz. Ann. Math. 126(3), 577–591 (1987)

    Article  MathSciNet  Google Scholar 

  11. Castryck, W., Denef, J., Vercauteren, F.: Computing zeta functions of nondegenerate curves, IMRP. Int. Math. Res. Pap. 72017, 57 (2006)

    Google Scholar 

  12. Coleff, N., Herrera, M.: Les courants résiduels associcés à une forme méromorphe, vol. 633. Lecture Notes in Mathematics. Springer, Berlin (1978)

  13. Cox D.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Cox D., Sidman J.: Secant varieties of toric varieties. J. Pure Appl. Algebra 209(3), 651–669 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Demailly, J.-P.: Complex analytic and algebraic geometry, Monograph. http://www-fourier.ujf-grenoble.fr/~demailly

  16. Dickenstein A., Sessa C.: Canonical representatives in moderate cohomology. Invent. Math. 80, 417–434 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ein L., Lazarsfeld R.: A geometric effective Nullstellensatz. Invent. Math. 137, 427–448 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)

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

    MathSciNet  MATH  Google Scholar 

  20. Hörmander L.: Generators for some rings of analytic functions. Bull. Am. Math. Soc. 73, 943–949 (1967)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Kollár J.: Effective Nullstellensatz for arbitrary ideals. J. Eur. Math. Soc. 1(3), 313–337 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lärkäng, R.: Residue currents associated with weakly holomorphic functions. arXiv:0910.3589 (preprint)

  25. Lazarsfeld R.: Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals. Springer, Berlin (2004)

    MATH  Google Scholar 

  26. Macaulay F.S.: The Algebraic Theory of Modular Systems. Cambridge University Press, Cambridge (1916)

    MATH  Google Scholar 

  27. Nöther M.: Über einen Satz aus der Theorie der algebraischen Functionen. Math. Ann. 6(3), 351–359 (1873)

    Article  MathSciNet  Google Scholar 

  28. Oda T.: Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 15. Springer, Berlin (1988)

  29. Passare M.: Residues, currents, and their relation to ideals of holomorphic functions. Math. Scand. 62(1), 75–152 (1988)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. Payne S.: Fujita’s very ampleness conjecture for singular toric varieties. Tohoku Math. J. 58(3), 447–459 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Skoda H.: Application des techniques L 2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. 5, 545–579 (1972)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Tuitman, J.: A mixed sparse effective nullstellensatz (2008, preprint)

  35. Weimann, M.: La trace en géométrie projective et torique. Ph.D. Thesis, Université Bordeaux (2006)

  36. Wulcan, E.: Some variants of Macaulay’s and Max Noether’s Theorems. J. Commut. Algebra (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Elizabeth Wulcan

Authors
  1. Elizabeth Wulcan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elizabeth Wulcan.

Additional information

E. Wulcan was partially supported by the Swedish Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wulcan, E. Sparse effective membership problems via residue currents. Math. Ann. 350, 661–682 (2011). https://doi.org/10.1007/s00208-010-0575-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0575-6

Keywords

Search

Navigation