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Faltings’s Version of Siegel’s Lemma

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1566)

Abstract

Siegel’s Lemma in its original form guarantees the existence of a small non-trivial integral solution of a system of linear equations with rational integer coefficients and with more variables than equations. It reads as follows: 1 Lemma. (C.L. Siegel) Let A = (a ij ) be an N × M matrix with rational integer coefficients. Put a = max i,j |a ij |. Tien, if N < M, the equation Ax = 0 has a solution x ∈ ℤM, x≠0, with

$$ \left\| x \right\| \leqslant (Ma)^{N/(M - N)} $$

where ‖ ‖ denotes the max-norm: ‖x‖=‖(x1,...,x M ‖=max1≤iM|x i | in ℝM.

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© 1993 Springer-Verlag Berlin Heidelberg

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Kooman, RJ. (1993). Faltings’s Version of Siegel’s Lemma. In: Edixhoven, B., Evertse, JH. (eds) Diophantine Approximation and Abelian Varieties. Lecture Notes in Mathematics, vol 1566. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48208-6_10

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  • DOI: https://doi.org/10.1007/978-3-540-48208-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57528-3

  • Online ISBN: 978-3-540-48208-6

  • eBook Packages: Springer Book Archive