Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762

 
 

 

Residues and effective Nullstellensatz


Authors: Carlos A. Berenstein and Alain Yger
Journal: Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 82-91
MSC (1991): Primary 14Q20; Secondary 13F20, 14C17, 32C30
DOI: https://doi.org/10.1090/S1079-6762-96-00011-X
MathSciNet review: 1412946
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathbf {K} $ be a commutative field; an algorithmic approach to residue symbols defined on a Noetherian $\mathbf {K} $-algebra $\mathbf {R} $ has been developed. It is used to prove an effective Nullstellensatz for polynomials defined over infinite factorial rings $\mathbf { A} $ equipped with a size. This result extends (and slightly improves) the previous work of the authors in the case $\mathbf { A} =\mathbf {Z} $.


References [Enhancements On Off] (What's this?)

  • [BGVY] C. A. Berenstein, R. Gay, A. Vidras, and A. Yger, Residue currents and Bézout identities, Progr. Math., vol. 114, Birkhäuser, Basel, 1993. MR 94m:32006
  • [BY1] C. A. Berenstein and A. Yger, Effective Bézout identities in $Q[z_{1},\dotsc ,z_{n}]$, Acta Math. 166 (1991), 69--120. MR 92f:32004
  • [BY2] ------, Residue calculus and effective Nullstellensatz, University of Maryland preprint.
  • [BGS] J. B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903--1027. MR 95j:14025
  • [B] D. W. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), 577--591. MR 89b:12001
  • [CGH] L. Caniglia, A. Galligo, and J. Heintz, Borne simple exponentielle pour les degrés dans le théorème des zéros de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 255--258. MR 90c:12002
  • [G] P. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976), 321--390. MR 55:8036
  • [GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 80b:14001
  • [H] R. Hartshorne, Residues and duality, Lect. Notes Math. 20, Springer, Berlin, 1966. MR 36:5145
  • [JKS] S. Ji, J. Kollár, and B. Shiffman, A global {\L}ojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992), 813--818. MR 92e:32007
  • [Ko] J. Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), 963--975. MR 89h:12008
  • [KK] M. Kreuzer and E. Kunz, Traces in strict Frobenius algebras and strict complete intersections, J. Reine Angew. Math. 381 (1987), 181--204. MR 89c:14022
  • [Ku] E. Kunz, Über den $n$-dimensionalen Residuensatz, Jahresber. Deutsch. Math.-Verein. 94 (1992), 170--188. MR 94a:13026
  • [Ky] A. M. Kytmanov, A transformation formula for Grothendieck residues and some of its applications, Siberian Math. J. 29 (1988), 495--499. MR 89j:32005
  • [L] J. Lipman, Residues and traces of differential forms via Hochschild homology, Contemp. Math., vol. 61, Amer. Math. Soc., Providence, 1987. MR 88b:14017
  • [LT] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97--116. MR 82f:14004
  • [M] H. Matsumura, Commutative algebra, Math. Lect. Notes, vol. 56, Benjamin/Cummings Pub. Co., Reading, Mass., 1980. MR 82i:13003
  • [Ph1] P. Philippon, Dénominateurs dans le théorème des zéros de Hilbert, Acta Arith. 58 (1991), 1--25. MR 92i:13008
  • [Ph2] ------, Sur des hauteurs alternatives, III, J. Math. Pures Appl. (9) 74 (1995), 345--365. CMP 95:15

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 14Q20, 13F20, 14C17, 32C30

Retrieve articles in all journals with MSC (1991): 14Q20, 13F20, 14C17, 32C30


Additional Information

Carlos A. Berenstein
Affiliation: Institute for Systems Research, University of Maryland, College Park, MD 20742
Email: carlos@src.umd.edu

Alain Yger
Affiliation: Laboratoire de Mathématiques Pures, Université Bordeaux Sciences, 33405 Talence, France
Email: yger@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/S1079-6762-96-00011-X
Keywords: Effective Nullstellensatz, residues, arithmetic B\'{e}zout theory
Received by editor(s): April 15, 1996
Additional Notes: This research has been partially supported by grants from NSA and NSF
Communicated by: Robert Lazarsfeld
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society