Diagonal equations over function fields. (English) Zbl 0612.10011

Let K be a function field in one variable over \({\mathbb{C}}\), and let S be a finite set of places of K. The main result of this paper (theorem 4) says that if \(u_ 1,...,u_ m\) are S-units which are linearly independent over \({\mathbb{C}}\) and satisfy \(u_ 1+...+u_ m=1\), then \[ \max \{\deg u_ 1,...,\deg u_ m\}\leq m(m-1)(2g-2+| S |). \] This estimate was discovered independently by W. D. Brownawell and D. W. Masser [Math. Proc. Camb. Philos. Soc. 100, 427-434 (1986; see the preceding review)], and improves a result of R. C. Mason [J. Number Theory 22, 190-207 (1986; Zbl 0578.10021)] in which the \({1/2}m(m-1)\) is replaced by \(4^{m-1}\). The author also gives a separate proof estimating the size of solutions to the diagonal equation \(\sum a_ i x^ n_ i=b\); and concludes that diagonal equations of this form have no non-trivial solutions if n is sufficiently large, generalizing a result of the reviewer [Trans. Am. Math. Soc. 273, 201-205 (1982; Zbl 0493.10020)]. However, it is clear that any estimate as above for the S- unit equation will give a result of this sort for diagonal equations, even ones of the form \(\sum a_ i x_ i^{n_ i}\). The language of the author’s proof differs from that of Brownawell, Masser, and Mason in that he works with the Weierstrass points of the function field K rather than working directly with Wronskians.
Reviewer: J.H.Silverman


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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