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Vanishing sums in function fields. (English) Zbl 0612.10010
Let \(k\) be a field of characteristic 0, and let \(F\) be a function field over \(k\) of genus \(g\). Normalize the valuations \(v\) on \(F\) to have value group the rational integers, and define the height of a point \(P=[u_ 1,\dots,u_ n]\) in projective space \({\mathbb P}^{n-1}(F)\) in the usual way, \(h(P)=\sum_{v}\max_{i}\{v(u_ i)\}\). The authors study solutions to the homogeneous \(n\)-variable ”unit” equation \[ d u_ 1+\dots+u_ n=0.\tag{*} \] A solution to (*) is called non-degenerate if every non-empty proper subset of \(\{u_ 1,\dots, u_ n\}\) is \(k\)-linearly independent. For \(p\geq 0\), let \(\gamma_ p=\max \{0, (p-1)(p-2)\}\); and for each valuation \(v\), let \(m(v)=m(v;u_ 1,\dots, u_ n)\) be the number of \(u_ i\)’s which are units at \(v\). Then the authors prove:
Theorem A: Any non- degenerate solution to (*) satisfies \[ h([u_ 1,\dots, u_ n])\leq \gamma_ n (2g-2)+\sum_{v}(\gamma_ n-\gamma_{m(v)}). \] As an immediate corollary, they obtain the estimate \[ h([u_ 1,\dots,u_ n])\leq (n-1)(n-2)\{| S| +2g-2\}, \tag{**} \] provided \(u_ 1,\dots ,u_ n\) are all \(S\)-units for some finite set \(S\).
The proof is a short and clever calculation using Wronskian determinants and elementary properties of derivations on function fields. The inequality (**) was independently discovered by J. F. Voloch [Bol. Soc. Bras. Mat. 16, No. 2, 29–39 (1985; Zbl 0612.10011)] who gave it a somewhat different proof based on the study of Weierstrass points. This generalized an earlier result of R. C. Mason [J. Number Theory 22, 190–207 (1986; Zbl 0578.10021)] who proved (**) with the \(1/2(n-1)(n-2)\) replaced by \(4^{n-2}\).
Reviewer: J.H.Silverman

MSC:
11D41 Higher degree equations; Fermat’s equation
11R58 Arithmetic theory of algebraic function fields
14H05 Algebraic functions and function fields in algebraic geometry
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References:
[1] Silverman, Math. Proc. Cambridge Philos. Soc 95 pp 3– (1984)
[2] Muir, A Treatise on the Theory of Determinants (1960)
[3] Cartan, Mathematica Cluj 7 pp 5– (1933)
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[6] DOI: 10.1016/0022-314X(86)90069-7 · Zbl 0578.10021 · doi:10.1016/0022-314X(86)90069-7
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