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79) But since the number of K's of degree n with h (K) = h is at most (n + 1) (2h + 1)n, the following assertion is obtained by applying Lemma 12 to the system of sets defined by (79). Lemma 16. For almost all real (complex) numbers co the system (77) has, for given w > n (w > (n - 1)/2) and for an arbitrary p0 > 0, at most finitely many algebraic numbers K e Kp0 as solutions. In other words, we are facing a nontrivial situation only if there are infinitely §8. THE EQUATION 02 = 1 33 many polynomials P satisfying the system (77) which have, if the height h (P) is sufficiently large, in addition to K E P at least one further root K ' E P sufficiently close to co .

Hence almost all points the square (47) belong to the set M2 . Finally we wish to state the well-known Borel-Cantelli Lemma, which plays an important part in metrical theorems of number theory, and to which we shall frequently refer later on. Lemma 12. Let A = U z°__ 1A i be a union of measurable linear or planar sets satisfying oc VmeaSA< 0o. i_1 (48) §4. INVARIANCE OF THE PARAMETERS wn(o) 23 Then the set of all those points which belong to infinitely many sets Ai has measure zero. Proof. Let B be the set of points belonging to infinitely many sets Aj.

P37 numbers r2, r3, , rk, are uniquely determined by the root K = K 1 of the polynomial P. We define the class KE (r) = K (r) to consist of all algebraic numbers K corresponding to a vector r. In the sequel we shall need the following relation concerning the quantities + r2, r3, , . rk. Lemma 23. If the class K (r) contains infinitely many elements, then k (i - 1 r; 1 n M j=2 (116) 1 2 Proof. Considering the discriminant D(P) of the polynomial P C P n with the root K = K 1 E K (r) and also the roots K 17 K 27 ' , K k, we obtain 1 < 1D(P)I = hen-2 ] 1Kt .

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