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Vanishing sums in function fields

Published online by Cambridge University Press:  24 October 2008

W. D. Brownawell
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA 16802, U.S.A.
D. W. Masser
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

Extract

Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height as

The sum formula on F shows that this is really a height on the projective space .

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

[1]Cartan, H.. Sur les zéros de combinaisons linéaires de p fonctions holomorphes données. Mathematica Cluj 7 (1933), 529;Google Scholar
Collected Works, vol. 1 (ed. Remmert, R. and Serre, J.-P.) (Springer-Verlag, 1979), 421445.Google Scholar
[2]Hayman, W. K.. Waring's Problem für analytische Funktionen. Bayerische Akademie der Wissenschaften, Math.-Nat. Klasse no. 1 (1984).Google Scholar
[3]Mason, R. C.. Diophantine Equations over Function Fields. London Math. Soc. Lecture Notes, vol. 96 (Cambridge University Press, 1984).CrossRefGoogle Scholar
[4]Mason, R. C.. Norm form equations. I. J. Number Theory 22 (1986), 190207 (see also similar titles II, III, IV, V).CrossRefGoogle Scholar
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[6]Silverman, J. H.. The S-unit equation over function fields. Math. Proc. Cambridge Philos. Soc. 95 (1984), 34.CrossRefGoogle Scholar
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