By Richard Bellman
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AUXILIARY CONSIDERATIONS 28 exist, for almost all complex numbers c), at most finitely many integer solutions a0, a j, a2, a3 with max (I ao 1, jail, < h of the inequality I I (66) I a0 + a1 w + a2(0 2 + a36) 31 < h-1 ah provided that 00 h-1 a2h < (67) 00. Proof. Let A be the set of those numbers cv for which the inequality (66) has infinitely many solutions and let c) E A be a fixed element. Then we select from the sequence of polynomials P (x) = a0 + a 1x + a 2x 2 + a 3x 3 satisfying (66) an infinite subsequence of polynomials PA, with the property that, letting Pk (x) _ h k 1Pk (x), lim pk(x) = p (x), (68) k->00 where p (x) = 80 +,8 1x + 02x2 + 03X 3 is any polynomial and convergence is defined with respect to the metric of four-dimensional euclidean space; in other words, the equation (68) is equivalent to the statement lim max T' - Aj I = 0, I k->0 i=0, 192v3 If K(/t), KS'), and K k) are the roots of the polynomial p A, (x) and if they are so numbered that min I - K1k)l (k = 1, 2, ...
1 > hP 1/2n holds because of (82). Therefore, it follows from the definition of w0 (see (83)) and from (2) that 4wo n+5 > 6 2n 6 4- n+ 5 6 n+5 4wn_1 n+ 1 6 Consequently, n4-1 (00 and thus x1 = ! (0o X21 < c (n, p) hp 6 §i. THE DOMAINS a ,(P) 1 37 _ n+1 x21 < c (/top) hp 6 xl (85) Analogously we obtain n+1 6 P) hp I (86) and furthermore, by means of (85) and (86), n+i xl (87) x3 1 < C (11, p) hp f Finally, an application of (85), (86), and (87) yields the inequality n± I I xj h x; i < c (it, p) h1' (88) 2 On the other hand, it is easy to see that the roots K 1, K 2, and K 3 must be nonreal if h > c (n, p).
6. REDUCTION TO POLYNOMIALS FROM Pn 25 5. REDUCTION TO IRREDUCIBLE POLYNOMIALS In the proof it turns out to be very important to be able to restrict the inequality (1) to the case where only irreducible polynomials (over the rational field) ti are admitted as solutions. We define wn (ce) as the supremum of the set of those numbers w for which the inequality (1) is satisfied by infinitely many irreducible polynomials of degree less than or equal to n. Then the following statement is true: Remark 2.
Analytic number theory: An introduction by Richard Bellman