ON THE ESTIMATES OF RESULTANT VALUES OF INTEGER POLYNOMIALS WITHOUT COMMON ROOTS

In the article we present an improvement to the lemma on the order of simultaneous zero approximation by the values of two integer polynomials without common roots from A. O. Gelfond’s monograph “Transcendental and algebraic numbers”. The lemma says that if two integer polynomials P1 and P2 of degree not exceeding n1 and n2 and of height not exceeding Qµ1 and Qµ2 respectively having no roots in common take values 1 Px Q 1( ) −τ < and 2 Px Q 2 ( ) −τ < at some transcendental point x ∈¡, then min( , ) τ τ < µ + µ +δ 1 2 12 21 n n . Gelfond’s lemma and similar results have important applications to many problems of the metric theory of Diophantine approximation. One of such applications is the result due to V. Bernik (1983) on the upper bound for the Hausdorff dimension of the set of real numbers with specified order of zero approximation by the values of integer polynomials. This result along with the result of A. Baker and W. Schmidt (1970) on the lower bound of the Hausdorff dimension of the set mentioned above gives the exact formula. In order to prove the upper bound V. Bernik improved and extended Gelfond’s lemma by considering the values of polynomials of degree not exceeding n and of height not exceeding Qµ on some interval of length Q−η and obtaining a stronger inequality τ+µ+ τ+µ−η < µ +δ 2max( , 0) 2 , n τ= τ τ min( , ). 1 2 However, the need to consider the same estimates for the degree and height of the polynomials is still restrictive and limits the range of problems this result could be applied to. In our work we consider the values of polynomials of different degrees and heights on an interval and obtain a stronger estimate by using higher order derivatives, thus improving and extending Gelfond’s lemma and existing similar results. The result is obtained using the methods of the theory of transcendental numbers.

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