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Abstract:

We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz. © 2013 Sociét. Mathématique de France. Tous droits réservé s.

Registro:

Documento: Artículo
Título:Heights of varieties in multiprojective spaces and arithmetic nullstellensätze
Autor:Dõandrea, C.; Krick, T.; Sombra, M.
Filiación:Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
IMAS, CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina
ICREA, Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Año:2013
Volumen:46
Número:4
Página de inicio:549
Página de fin:627
Título revista:Annales Scientifiques de l'Ecole Normale Superieure
Título revista abreviado:Ann. Sci. Ec. Norm. Super.
ISSN:00129593
CODEN:ASENA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00129593_v46_n4_p549_Doandrea

Referencias:

  • Aschenbrenner, M., Ideal membership in polynomial rings over the integers (2004) Journal of the American Mathematical Society, 17 (2), pp. 407-441. , DOI 10.1090/S0894-0347-04-00451-5, PII S0894034704004515
  • Berenstein, C.A., Yger, A., Effective B.Zout identities in ℚ[z1; Zn] (1991) Acta Math, 166, pp. 69-120
  • Bilu, Y.F., Strambi, M., Quantitative Riemann existence theorem over a number field (2010) Acta Arith, 145, pp. 319-339
  • Bombieri, E., Bourgain, J., Konyagin, S.V., Roots of polynomials in subgroups of F* p and applications to congruences (2009) Int. Math. Res. Not, (2009), pp. 802-834
  • Brownawell, W.D., Bounds for the degrees in the Nullstellensatz (1987) Ann. of Math, 126, pp. 577-591
  • Brownawell, W.D., The Hilbert Nullstellensatz, inequalities for polynomials, and algebraic independence (2001) Introduction to Algebraic Independence Theory, 1752, pp. 239-248. , Lecture Notes in Math, Springer
  • Burgos Gil, J.I., Philippon, P., Sombra, M., Arithmetic geometry of toric varieties Metrics, Measures and Heights, , preprint arXiv: 1105.5584
  • Chow, W.-L., Van Derwaerden, B.L., Zur algebraischen Geometrie. IX (1937) Math. Ann, 113, pp. 692-704
  • Dahan, X., Kadri, A., Schost, E., Bit-size estimates for triangular sets in positive dimension (2012) J. Complexity, 28, pp. 109-135
  • David, S., Philippon, P., Minorations des hauteurs normalis.es des sous-vari.t.s des tores (1999) Ann. Scuola Norm. Sup. Pisa Cl. Sci, 28, pp. 489-543
  • Fulton, W., Intersection theory (1984) Ergebn. Math. Grenzg, 2. , Springer
  • Gelofand, I.M., Kapranov, M.M., Zelevinsky, A.V., (1994) Discriminants, Resultants, and Multidimensional Determinants, , Mathematics: Theory & Applications Birkhäuser
  • Hartshorne, R., Algebraic geometry (1977) Graduate Texts in Math, 52. , Springer
  • Jelonek, Z., On the effective Nullstellensatz (2005) Inventiones Mathematicae, 162 (1), pp. 1-17. , DOI 10.1007/s00222-004-0434-8
  • Jouanolou, J.-P., Théorèmes de bertini et applicationss (1983) Progress in Math, 42. , Birkhäuser
  • Koiran, P., Hilbert's Nullstellensatz is in the polynomial hierarchy (1996) Journal of Complexity, 12 (4), pp. 273-286. , DOI 10.1006/jcom.1996.0019
  • Kresch, A., Tschinkel, Y., Effectivity of brauer-manin obstructions (2008) Adv. Math, 218, pp. 1-27
  • Krick, T., Pardo, L.M., A computational method for Diophantine approximation (1996) Algorithms in Algebraic Geometry and Applications (Santander, 1994, 143, pp. 193-253. , Progr. Math, Birkhäuser
  • Krick, T., Pardo, L.M., Sombra, M., Sharp estimates for the arithmetic Nullstellen-satz (2001) Duke Math. J, 109, pp. 521-598
  • Lang, S., (1983) Fundamentals of Diophantine Geometry, , Springer
  • Lang, S., (1993) Algebra, , third ed., Addison-Wesley Publishing Co., Inc., Reading, Mass
  • Lelong, P., Mesure de Mahler et calcul de constantes universelles pour les polynômes de n variables (1994) Math. Ann, 299, pp. 673-695
  • Macaulay, F.S., Some formulae in elimination (1902) Proc. London Math. Soc, 1, pp. 3-27
  • Maillot, V., Géométrie d'Arakelov des variétés toriques et fibrés en droites intégrables (2000) M.m. Soc. Math. France, 80
  • Pedersen, P., Sturmfels, B., Product formulas for resultants and chow forms (1993) Math. Z, 214, pp. 377-396
  • Perron, O., (1951) Algebra. I. Die Grundlagen, , Walter de Gruyter & Co
  • Philippon, P., Critères pour l'indépendance alg.brique (1986) Publ. Math. I.H.É.S, 64, pp. 5-52
  • Philippon, P., D.nominateurs dans le théorème des z.ros de hilbert (1991) Acta Arith, 58, pp. 1-25
  • Philippon, P., Sur des hauteurs alternatives. I (1991) Math. Ann, 289, pp. 255-283
  • Philippon, P., Sur des hauteurs alternatives. III (1995) J. Math. Pures Appl, 74, pp. 345-365
  • Philippon, P., Sombra, M., Hauteur normalisée des variétés toriques projectives (2008) J. Inst. Math. Jussieu, 7, pp. 327-373
  • Rémond, G., Limination multihomogène (2001) Introduction to Algebraic Independence Theory, 1752, pp. 53-81. , Lecture Notes in Math, Springer
  • Rémond, G., G.om.trie diophantienne multiprojective (2001) Introduction to Algebraic Independence Theory, 1752, pp. 95-131. , Lecture Notes in Math, Springer
  • Rémond, G., Nombre de points rationnels des courbes (2010) Proc. Lond. Math. Soc, 101, pp. 759-794
  • Smietanski, F., A parametrized nullstellensatz (1993) Computational Algebraic Geometry (Nice, 1992, 109, pp. 287-300. , Progr. Math, Birkhäuser
  • Smyth, C.J., A Kronecker-type theorem for complex polynomials in several variables (1981) Canad. Math. Bull, 24, pp. 447-452
  • Sombra, M., The height of the mixed sparse resultant (2004) American Journal of Mathematics, 126 (6), pp. 1253-1260
  • Teissier, B., Résultats Récents d'Algèbre Commutative Effective, 1989 (90). , Séminaire Bourbaki, exposé no 718
  • (1990) Astérisque, 189-190, pp. 107-131

Citas:

---------- APA ----------
Dõandrea, C., Krick, T. & Sombra, M. (2013) . Heights of varieties in multiprojective spaces and arithmetic nullstellensätze. Annales Scientifiques de l'Ecole Normale Superieure, 46(4), 549-627.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00129593_v46_n4_p549_Doandrea [ ]
---------- CHICAGO ----------
Dõandrea, C., Krick, T., Sombra, M. "Heights of varieties in multiprojective spaces and arithmetic nullstellensätze" . Annales Scientifiques de l'Ecole Normale Superieure 46, no. 4 (2013) : 549-627.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00129593_v46_n4_p549_Doandrea [ ]
---------- MLA ----------
Dõandrea, C., Krick, T., Sombra, M. "Heights of varieties in multiprojective spaces and arithmetic nullstellensätze" . Annales Scientifiques de l'Ecole Normale Superieure, vol. 46, no. 4, 2013, pp. 549-627.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00129593_v46_n4_p549_Doandrea [ ]
---------- VANCOUVER ----------
Dõandrea, C., Krick, T., Sombra, M. Heights of varieties in multiprojective spaces and arithmetic nullstellensätze. Ann. Sci. Ec. Norm. Super. 2013;46(4):549-627.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00129593_v46_n4_p549_Doandrea [ ]