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DION 2005 The role of complex conjugation in transcendental number theory Michel Waldschmidt

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DION 2005

The role of complex conjugation in transcendental number theory

Michel Waldschmidt

Abstract.

In his two well known 1968 papers ”Contributions to the theory of transcen- dental numbers”, K. Ramachandra proved several results showing that, in certain explicit sets {x

1

, . . . , x

n

} of complex numbers, one element at least is transcendental. In specific cases the number n of elements in the set was 2 and the two numbers x

1

, x

2

were both real. He then noticed that the conclusion is equivalent to say that the complex number x

1

+ ix

2

is transcendental.

In his 2004 paper published in the Journal de Th´ eorie des Nombres de Bordeaux, G. Diaz investigates how complex conjugation can be used for the transcendence study of the values of the exponential function. For instance, if log α

1

and log α

2

are two nonzero logarithms of algebraic numbers, one of them being either real of purely imaginary, and not the other, then the product (log α

1

)(log α

2

) is transcendental.

We survey Diaz results and produce further similar ones.

Michel Waldschmidt,

Universit´ e P. et M. Curie (Paris VI),

Institut de Math´ ematiques de Jussieu, UMR 7586 CNRS, Probl` emes Diophantiens, Case 247,

175, rue du Chevaleret F-75013 Paris, France.

e-mail: [email protected] http://www.math.jussieu.fr/

∼

miw/

1

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