The generalized Fermat equation in function fields
Let \(k\) be an algebraically closed field of characteristic 0 and \(K=k(t)\) a function field over \(k\). Let \(a_ i\) (\(i=1,\dots,n\)) be fixed elements in \(K\) (not all zero) and consider the solutions \(z_ 1,\dots,z_ n\in K\) of the generalized Fermat equation \[ a_ 1z^ r_ 1+\dots+a_ nz^ r_ n=0. \tag{1} \] In case of the basic Fermat equation (\(n=3\)) \textit{J. H. Silverman} [Trans. Am. Math. Soc. 273, 201-205 (1982; Zbl 0493.10020)] and \textit{J. F. Voloch} [Bol. Soc. Bras. Mat. 16, No. 2, 29-39 (1985; Zbl 0612.10011)] gave upper bounds for the height and the number of solutions of the equation as a function of \(r\) and the heights of the coefficients \(a_ i\). On the other hand \textit{J. Mueller} [Compos. Math. 73, 189-197 (1990; Zbl 0697.10015)] proved, that if \(r>30\), then a ``non- degenerate'' equation \(ax^ r+by^ r=c\) can have at most two non-zero solutions \(x,y\in K\). This bound is best possible and independent of the coefficients \(a,b,c\in K\). In the present paper the authors extend the above mentioned result of J. Mueller to the generalized Fermat equation (1), namely they prove, that if \(r>n!(n!-2)\), then the solution of (1) forms at most \(n!^{n!}\) families of solutions, where the families have an explicitly given simple structure. The main tool of the proof is the function field analogue of the \(abc\)-inequality.
scientific article; zbMATH DE number 23713
scientific article