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
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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 $\mathbf {A} =\mathbf {Z}$.
- Carlos A. Berenstein, Roger Gay, Alekos Vidras, and Alain Yger, Residue currents and Bezout identities, Progress in Mathematics, vol. 114, Birkhäuser Verlag, Basel, 1993. MR 1249478
- Carlos A. Berenstein and Alain Yger, Effective Bezout identities in ${\bf Q}[z_1,\cdots ,z_n]$, Acta Math. 166 (1991), no. 1-2, 69–120. MR 1088983, DOI https://doi.org/10.1007/BF02398884
- ---, Residue calculus and effective Nullstellensatz, University of Maryland preprint.
- J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), no. 4, 903–1027. MR 1260106, DOI https://doi.org/10.1090/S0894-0347-1994-1260106-X
- W. Dale Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), no. 3, 577–591. MR 916719, DOI https://doi.org/10.2307/1971361
- Léandro Caniglia, André Galligo, and Joos Heintz, Borne simple exponentielle pour les degrés dans le théorème des zéros sur un corps de caractéristique quelconque, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 6, 255–258 (French, with English summary). MR 956817
- Phillip A. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390. MR 435074, DOI https://doi.org/10.1007/BF01390145
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
- Shanyu Ji, János Kollár, and Bernard Shiffman, A global Łojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992), no. 2, 813–818. MR 1046016, DOI https://doi.org/10.1090/S0002-9947-1992-1046016-6
- János Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975. MR 944576, DOI https://doi.org/10.1090/S0894-0347-1988-0944576-7
- Martin Kreuzer and Ernst Kunz, Traces in strict Frobenius algebras and strict complete intersections, J. Reine Angew. Math. 381 (1987), 181–204. MR 918848
- E. Kunz, Über den $n$-dimensionalen Residuensatz, Jahresber. Deutsch. Math.-Verein. 94 (1992), no. 4, 170–188 (German). MR 1190210
- A. M. Kytmanov, A formula for the transformation of the Grothendieck residue and some of its applications, Sibirsk. Mat. Zh. 29 (1988), no. 3, 198–202, 223 (Russian); English transl., Siberian Math. J. 29 (1988), no. 3, 495–499 (1989). MR 953040, DOI https://doi.org/10.1007/BF00969664
- Joseph Lipman, Residues and traces of differential forms via Hochschild homology, Contemporary Mathematics, vol. 61, American Mathematical Society, Providence, RI, 1987. MR 868864
- Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Patrice Philippon, Dénominateurs dans le théorème des zéros de Hilbert, Acta Arith. 58 (1991), no. 1, 1–25 (French). MR 1111087, DOI https://doi.org/10.4064/aa-58-1-1-25
- ---, Sur des hauteurs alternatives, III, J. Math. Pures Appl. (9) 74 (1995), 345–365.
- C. A. Berenstein, R. Gay, A. Vidras, and A. Yger, Residue currents and Bézout identities, Progr. Math., vol. 114, Birkhäuser, Basel, 1993.
- C. A. Berenstein and A. Yger, Effective Bézout identities in $Q[z_{1},\dotsc ,z_{n}]$, Acta Math. 166 (1991), 69–120.
- ---, Residue calculus and effective Nullstellensatz, University of Maryland preprint.
- J. B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027.
- D. W. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), 577–591.
- 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.
- P. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390.
- P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978.
- R. Hartshorne, Residues and duality, Lect. Notes Math. 20, Springer, Berlin, 1966.
- S. Ji, J. Kollár, and B. Shiffman, A global Łojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992), 813–818.
- J. Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), 963–975.
- M. Kreuzer and E. Kunz, Traces in strict Frobenius algebras and strict complete intersections, J. Reine Angew. Math. 381 (1987), 181–204.
- E. Kunz, Über den $n$-dimensionalen Residuensatz, Jahresber. Deutsch. Math.-Verein. 94 (1992), 170–188.
- A. M. Kytmanov, A transformation formula for Grothendieck residues and some of its applications, Siberian Math. J. 29 (1988), 495–499.
- J. Lipman, Residues and traces of differential forms via Hochschild homology, Contemp. Math., vol. 61, Amer. Math. Soc., Providence, 1987.
- 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.
- H. Matsumura, Commutative algebra, Math. Lect. Notes, vol. 56, Benjamin/Cummings Pub. Co., Reading, Mass., 1980.
- P. Philippon, Dénominateurs dans le théorème des zéros de Hilbert, Acta Arith. 58 (1991), 1–25.
- ---, Sur des hauteurs alternatives, III, J. Math. Pures Appl. (9) 74 (1995), 345–365.
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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
Keywords:
Effective Nullstellensatz,
residues,
arithmetic Bézout 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
