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 \(u_1,\ldots,u_m\) are \(S\)-units which are linearly independent over \(\mathbb{C}\) and satisfy \(u_1+\cdots+u_m=1\), then \[ \max \{\deg u_1,\ldots,\deg u_m\} \leq m(m-1)(2g - 2 + | S |). \] This estimate was discovered independently by W. D. Brownawell and D. W. Masser [Math. Proc. Camb. Philos. Soc. 100, 427–434 (1986; Zbl 0612.10010)] (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 \(\tfrac12 m(m-1)\) 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.