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This paper is devoted to the establishment of explicit bounds on the rational function solutions of a general class of equations in several variables. The first general result on Diophantine equations over function fields was discovered in 1930 [1], as an analogy on the work of Thue on number fields: it was shown that the degrees of polynomial solutions X, Y in k[z] of f(X, Y) = c are bound for each c: here f denotes an irreducible binary form over k[z] of degree at least three, and k is an algebraically closed field of characteristic zero. The Manin-Grauert theorem [7] extended this conclusion to the rational function solutions X, Y in k(z) of any equation in two variables over k(z), provided that the curve corresponding to the equation has genus two or more: this is the analogue for function fields of the Mordell conjecture for number fields, proved by Faltings in 1983. Parsin [6] made the Manin-Grauert theorem effective by furnishing explicit bounds on the degrees of X and Y: this approach followed Grauert and Shafarevitch in relying heavily on algebraic geometry. In 1976 a different attack was made by Schmidt [8] using the theory of algebraic differential equations, first developed by Kolchin and Osgood. This method produced very good bounds for equations such as the Thue equation discussed above: for example, Schmidt provides the bound

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