English

18 Pages

Gain access to the library to view online

__
Learn more
__

Description

Niveau: Supérieur, Doctorat, Bac+8

On the values of a class of analytic functions at algebraic points Boris ADAMCZEWSKI, Yann BUGEAUD and Florian LUCA Abstract. We introduce a class of analytic functions of number theo- retic interest, namely stammering functions. It has been recently proved that these functions take transcendental values at some algebraic points. In the present paper, we establish a general transcendence criterion that extends these results. Another aim is to underline the main difficulties arising from the use of the Schmidt Subspace Theorem in this context. 1. Introduction In 1844, Liouville established that the real number ∑ k≥1 10?k! is transcendental. With a similar method, it is nowadays an easy exercise to extend his result as follows (see [8], page 2). Theorem L. Let ? be an algebraic number with 0 < |?| < 1. Then, the complex number∑ k≥1 ?k! is transcendental. In other words, the analytic function f : z 7? ∑ k≥1 zk! takes transcendental values at every nonzero algebraic point in its open disc of convergence. A similar result was obtained by Mahler [6] for a much wider class of functions f , including some classical series such as F(z) = ∑ k≥0 z2 k or the Thue–Morse function T (z) = ∑ k≥0 tkzk, where tk = 1 (resp.

On the values of a class of analytic functions at algebraic points Boris ADAMCZEWSKI, Yann BUGEAUD and Florian LUCA Abstract. We introduce a class of analytic functions of number theo- retic interest, namely stammering functions. It has been recently proved that these functions take transcendental values at some algebraic points. In the present paper, we establish a general transcendence criterion that extends these results. Another aim is to underline the main difficulties arising from the use of the Schmidt Subspace Theorem in this context. 1. Introduction In 1844, Liouville established that the real number ∑ k≥1 10?k! is transcendental. With a similar method, it is nowadays an easy exercise to extend his result as follows (see [8], page 2). Theorem L. Let ? be an algebraic number with 0 < |?| < 1. Then, the complex number∑ k≥1 ?k! is transcendental. In other words, the analytic function f : z 7? ∑ k≥1 zk! takes transcendental values at every nonzero algebraic point in its open disc of convergence. A similar result was obtained by Mahler [6] for a much wider class of functions f , including some classical series such as F(z) = ∑ k≥0 z2 k or the Thue–Morse function T (z) = ∑ k≥0 tkzk, where tk = 1 (resp.

- salem numbers
- schmidt subspace
- complex algebraic
- mahler's method
- every nonnegative integer
- sequences generated
- sequences
- integer such

Subjects

Informations

Published by | mijec |

Reads | 19 |

Language | English |

Report a problem