×

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
PDFBibTeX XMLCite
Full Text: DOI

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)
[4] Mason, Diophantine Equations over Function Fields 96 (1984) · Zbl 0533.10012 · doi:10.1017/CBO9780511752490
[5] Remmert, Collected Works 1 pp 421– (1979)
[6] DOI: 10.1016/0022-314X(86)90069-7 · Zbl 0578.10021 · doi:10.1016/0022-314X(86)90069-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.