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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:
  Silverman, Math. Proc. Cambridge Philos. Soc 95 pp 3– (1984)  Muir, A Treatise on the Theory of Determinants (1960)  Cartan, Mathematica Cluj 7 pp 5– (1933)  Mason, Diophantine Equations over Function Fields 96 (1984) · Zbl 0533.10012 · doi:10.1017/CBO9780511752490  Remmert, Collected Works 1 pp 421– (1979)  DOI: 10.1016/0022-314X(86)90069-7 · Zbl 0578.10021 · doi:10.1016/0022-314X(86)90069-7
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