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Construction of rational functions on a curve

Published online by Cambridge University Press:  24 October 2008

J. Coates
J. Coates
Affiliation:
Department of Pure Mathematics, Cambridge
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Extract

1. Introduction. In the study of diophantine equations in two variables, it is often necessary to consider rational functions on a curve with prescribed zeros and poles. Although it is well known that such functions can, in principle, always be effectively constructed, the detailed proof does not appear to have been given. The purpose of the present paper is to give the complete proof of such a construction. Our method, and the statement of our results, are motivated by the applications to diophantine equations which we have in mind. In particular, our results will play an important role in a subsequent paper (1), in which explicit bounds will be established for the integer points on any curve of genus 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 105 - 123
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Baker, A. and Coates, J.Integer points on curves of genus 1. Proc. Cambridge Philos. Soc. 67 (1970), 595.CrossRefGoogle Scholar
(2)Hasse, H.Zahlentheorie (Berlin, 1949).Google Scholar
(3)Hensel, K. and Landsberg, G.Theorie der algebraischen Funktionen einer variablen (Leipzig, 1902).Google Scholar
(4)Van der, Waerden, Algebra, B. Modern, vol. I (New York, 1953).Google Scholar