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Vanishing sums in function fields

  • W. D. Brownawell (a1) and D. W. Masser (a2)


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 .



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[1]Cartan, H.. Sur les zéros de combinaisons linéaires de p fonctions holomorphes données. Mathematica Cluj 7 (1933), 529;
Collected Works, vol. 1 (ed. Remmert, R. and Serre, J.-P.) (Springer-Verlag, 1979), 421445.
[2]Hayman, W. K.. Waring's Problem für analytische Funktionen. Bayerische Akademie der Wissenschaften, Math.-Nat. Klasse no. 1 (1984).
[3]Mason, R. C.. Diophantine Equations over Function Fields. London Math. Soc. Lecture Notes, vol. 96 (Cambridge University Press, 1984).
[4]Mason, R. C.. Norm form equations. I. J. Number Theory 22 (1986), 190207 (see also similar titles II, III, IV, V).
[5]Muir, T. and Metzler, W. H.. A Treatise on the Theory of Determinants (Dover, 1960).
[6]Silverman, J. H.. The S-unit equation over function fields. Math. Proc. Cambridge Philos. Soc. 95 (1984), 34.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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