Diophantine Approximation and Transcendence Theory Seminar, Bonn (FR
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1290 G. Wustholz (Ed.)
Diophantine Approximation and Transcendence Theory Seminar, Bonn (FRG) May - June 1985
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Editor
Gisbert Wustholz Mathematik, ETH Zentrum 8092 Zurich, Switzerland
Mathematics Subject Classification (1980): 10B10, 10B45, 10F05, 10F35, 10F37, 12C 15, 14D99, 14K99 ISBN 3-540-18597-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18597-6 Springer-Verlag New York Berlin Heidelberg
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INTRODUCTION
In 1985 the traditional Arbeitstagung at Bonn was cancelled. Instead a number of workshops were organized during Spring and Summer 1985 by various organizers. One of these workshops was on number theory with special emphasis on diophantine problems and transcendence. It took place at the Max-Planck-Institut fur Mathematik at Bonn in May - June 1985. A great number of leading mathematicians in the subject were invited for a certain period to discuss mathematics and problems. It seems that a very fruitful atmosphere was created which is reflected by quite a number of joint papers which were written during this time at Bonn or at least initiated there. We are very happy to present in this volume a selection of papers that grew out of this workshop at Bonn. It consists entirely of research papers which cover various important aspects of the field and each of them presents a fundamental contribution to the subject. In the first contribution by Colliot-Thelene, Kanevsky and Sansuc an effective algorithm for the calculation of the Manin obstruction for the Hasse principle for diagonal cubic surfaces is given. Then in the next article by Masser on small values of heights on families of abelian varieties, an effective lower bound for the variation of the Neron- Tate height in families of abelian varieties is given. In two further papers, one by Brownawell and another by Brownawell and Tubbs, questions on large transcendence degree are studied. The authors obtain very precise lower bounds for the transcendence degree of fields generated by values of elliptic or exponential functions. In the next two contributions multiplicity estimates for group varieties are studied in the most simplest case. Here extremely sharp results are given and applied to Baker's theory of linear forms in logarithms. In the last paper, by E. Bombieri, the number of solution
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