ON THE ORDER OF ZERO APPROXIMATION BY IRREDUCIBLE DIVISORS OF INTEGER POLYNOMIALS

Abstract

In the article we present an improvement to the lemma on the order of zero approximation by irreducible divisors of integer polynomials from A. O. Gelfond’s monograph “Transcendental and algebraic numbers”. The lemma says that if a polynomial P(x)∈Z[x] of degree not exceeding n and of height not exceeding Q satisfies inequality P(x) < Q−w, w > 6n, for some transcendental point x∈Z, then there exists a divisor d(x)∈Z[x] of P(x) that can be written as a degree of some polynomial irreducible over the field of rational numbers satisfying d(x) < Q−w+6n. 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 of 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 using a weaker condition w > 3n and obtaining a better estimate d(x) < Q−w+n, as well as by considering the values of polynomials on an interval. However, the condition on w is still restrictive and limits the range of problems this result could be applied to. In our work, we improve the existing results by obtaining the estimate d(x) < Q−w+n−1 on some interval for any w. The result is obtained using the methods of the theory of transcendental numbers.