E. Bombieri, A.J. van der Poorten and J. D. Vaaler. Effective Measures of Irrationality for Cubic Extensions of Number Fields, Ann Scuola Norm. Sup. Pisa Cl. Sci. (to appear). Home/Search Document Not in Database Summary Related Articles Check |
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On The Distribution Of Points In Projective Space Of Bounded Height - Choi (Correct)
.... ; ff N Gamma1 ] for the homogeneous coordinates of a generic element in P N Gamma1 (k v ) We let the quotient map OE : k N v Gamma f0g Gamma P N Gamma1 (k v ) be OE(ff 0 ; ff 1 ; Delta Delta Delta ; ff N Gamma1 ) ff 0 ; ff 1 ; Delta Delta Delta ; ff N Gamma1 ] As in [1], 5] and [9] one can define a projective metric on P N Gamma1 (k v ) as follows. If ff and fi belong to P N Gamma1 (k v ) then we define Delta v (ff; fi) kff fik v kffk v kfik v (1.3) and ffi v (ff; fi) jff fij v jffj v jfij v where is the wedge product. It follows from ....
....Y v jfij v ; 1.8) where the product is over all places v of k. In view of the product formula, the height function is well defined on P N Gamma1 (k) To illustrate a basic Diophantine inequality in this setting, we state the following projective form of Dirichlet s Theorem (e.g. Theorem 1 of [1] or Theorem 1 of [5] Dirichlet s Theorem. Let ff belong to P N Gamma1 (k v ) belong to k v and assume that 1 j j v . Then there exists fi in P N Gamma1 (k) such that (i) H(fi) c k (N)j j N Gamma1 v , ii) ffi v (ff; fi) c k (N)fj j v H(fi)g Gamma1 , where c k (N) ....
E. Bombieri, A.J. van der Poorten and J. D. Vaaler. Effective Measures of Irrationality for Cubic Extensions of Number Fields, Ann Scuola Norm. Sup. Pisa Cl. Sci. (to appear).
Diophantine Approximation in Projective Space - Choi, Vaaler Self-citation (Vaaler) (Correct)
....a point fi in k such that the height of fi is bounded by a suitable parameter and jff Gamma fij v is relatively small. And for special numbers ff it is a basic problem of Diophantine approximation to show that jff Gamma fij v cannot be too small if the height of fi is bounded. In a recent paper [2] such problems were reformulated in projective space over k v by replacing the Omega Gammae metric determined by j j v with a projective metric ffi v . Our purpose here is to give a proof of the projective form of Dirichlet s Theorem and to prove a useful inequality for the projective metric. We ....
....purpose here is to give a proof of the projective form of Dirichlet s Theorem and to prove a useful inequality for the projective metric. We also discuss some open problems suggested by these results. At each place v of k we use two absolute values j j v and k k v which are determined as in [1] [2], or [3] Thus we have jxj v = kxk dv=d v for all x in k v , where d = k : Q] and d v = k v : Q v ] These absolute values have unique extensions to Omega v , the completion of an algebraic closure of k v . We extend j j v to a norm on finite dimensional vector spaces over Omega v as ....
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E. Bombieri, A.J. van der Poorten, and J.D. Vaaler, Effective measures of irrationality for cubic extensions of number fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 211--248.
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