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A Generalization of the Subspace Theorem With Polynomials of Higher Degree

  • Jan-Hendrik Evertse
  • Roberto G. Ferretti
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

1.1 The Subspace Theorem can be stated as follows. Let K be a number field (assumed to be contained in some given algebraic closure Open image in new window of ℚ), n a positive integer, 0 < δ 1 and S a finite set of places of K. For v ∈ S, let \( L_0^{\left( v \right)} , \ldots ,L_n^{\left( v \right)} \) be linearly independent linear forms in Open image in new window [x 0,...,x n ]. Then the set of solutions x ∈ℙn(K) of
$$ \log \left( {\prod\limits_{v \in S} {\prod\limits_{i = 0}^n {\frac{{\left| {L_i^{\left( v \right)} \left( x \right)} \right|_v }} {{\left\| x \right\|_v }}} } } \right) \leqslant - \left( {n + 1 + \delta } \right)h\left( x \right) $$
(1.1)
is contained in the union of finitely many proper linear subspaces of ℙn.

Keywords

Diophantine approximation subspace theorem 

2000 Mathematics subject classification

11J68 11J25 

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References

  1. 1.
    Chardin, M.: Une majoration de la fonction de Hubert et ses conséquences pour l’interpolation algébrique. Bull. Soc. Math. F. 117, 305–318 (1989)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Corvaja, P., Zannier, U.: On a general Thue’s equation. Am. J. Math. 126, 1033–1055 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Evertse, J.-H.: On equations in S-units and the Thue-Mahler equation. Invent. Math. 75, 561–584 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Evertse, J.-H., Ferretti, R.G.: Diophantine inequalities on projective varieties. Int. Math. Res. Not. 2002, 1295–1330 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Evertse, J.-H., Schlickewei, H.P.: A quantitative version of the absolute subspace theorem. J. Reine Angew. Math. 548, 21–127 (2002)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Faltings, G., Wüstholz, G.: Diophantine approximations on projective spaces. Invent. Math. 116, 109–138 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ferretti, R.G.: Mumford’s degree of contact and Diophantine approximations. Compos. Math. 121, 247–262 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ferretti, R.G.: Diophantine approximations and toric deformations. Duke Math. J. 118, 493–522 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. II, Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar
  10. 10.
    Mumford, D.: Stability of projective varieties. Enseign. Math. II Sér. 23, 39–110 (1977)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Rémond, G.: Élimination multihomogène. In: Nesterenko, Yu.V., Philippon, P. (eds.) Introduction to Algebraic Independence Theory. Lect. Notes Math., vol. 1752, pp. 53–81. Springer, Heidelberg (2001)Google Scholar
  12. 12.
    Rémond, G.: Géométrie diophantienne multiprojective. In: Nesterenko, Yu.V., Philippon, P. (eds.) Introduction to Algebraic Independence Theory. Lect. Notes Math., vol. 1752, pp. 95–131. Springer, Heidelberg (2001)Google Scholar
  13. 13.
    Schlickewei, H.P.: The p-adic Thue-Siegel-Roth-Schmidt theorem. Arch. Math. 29, 267–270 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Schmidt, W.M.: Norm form equations. Ann. Math. 96, 526–551 (1972)CrossRefGoogle Scholar
  15. 15.
    Schmidt, W.M.: Simultaneous approximation to algebraic numbers by elements of a number field. Monatsh. Math. 79, 55–66 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schmidt, W.M.: The subspace theorem in diophantine approximation. Compos. Math. 96, 121–173 (1989)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jan-Hendrik Evertse
    • 1
  • Roberto G. Ferretti
    • 2
  1. 1.Mathematisch InstituutUniversiteit LeidenLeidenThe Netherlands
  2. 2.Università della Svizzera ItalianaLuganoSwitzerland

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