On approximation to real numbers by algebraic numbers

Tishchenko, K. I.

doi:10.4064/aa-94-1-1-24

Cited by 3 publications

(3 citation statements)

References 4 publications

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“…The first part of this lemma was already proved in [10] (see Lemma 3.4). To obtain (2.3), we put G 1 (x) = g (1)…”

Section: Auxiliary Lemmasmentioning

confidence: 82%

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On approximation of real numbers by algebraic numbers of bounded degree

Tsishchanka

2007

Journal of Number Theory

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“…The first part of this lemma was already proved in [10] (see Lemma 3.4). To obtain (2.3), we put G 1 (x) = g (1)…”

Section: Auxiliary Lemmasmentioning

confidence: 82%

“…The first announcement of it was made in [9]. The paper [10] contains the most comprehensive description of the method. The idea can be interpreted in the following way.…”

Section: Introductionmentioning

confidence: 99%

On approximation of real numbers by algebraic numbers of bounded degree

Tsishchanka

2007

Journal of Number Theory

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“…For proofs of Theorem 4.5 and related results, the reader may consult Chapter 3 of [15]. Slight improvements on (4.1) have been subsequently obtained by Tishchenko [77,78], with very technical proofs. Davenport and Schmidt [31] have given uniform upper bounds for the exponentsλ n andŵ n .…”

Section: Bounds For the Exponentsmentioning

confidence: 99%

Exponents of Diophantine Approximation

Bugeaud¹,

Laurent²

2016

Dynamics and Analytic Number Theory

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Counting real algebraic numbers with bounded derivative of minimal polynomial

Kudin

Vasilyev

2019

Int. J. Number Theory

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In this paper we consider the problem of counting algebraic numbers α of fixed degree n and bounded height Q such that the derivative of the minimal polynomial Pα(x) of α is bounded,This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of α defined above on the interval − 1 2 , 1 2 doesn't exceed c1(n)Q n+1− 1 7 v for Q > Q0(n) and 1.4 ≤ v ≤ 7 16 (n + 1). Our result is based on an improvement to the lemma on the order of zero approximation by irreducible divisors of integer polynomials from A. Gelfond's monograph "Transcendental and algebraic numbers". The improvement provides a stronger estimate for the absolute value of the divisor in real points which are located far enough from all algebraic numbers of bounded degree and height and it's based on the representation of the resultant of two polynomials as the determinant of Sylvester matrix for the shifted polynomials.

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On approximation to real numbers by algebraic numbers

Cited by 3 publications

References 4 publications

On approximation of real numbers by algebraic numbers of bounded degree

On approximation of real numbers by algebraic numbers of bounded degree

Exponents of Diophantine Approximation

Counting real algebraic numbers with bounded derivative of minimal polynomial

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