Equations \(a^ x=R(x,y)\)
Let \(\alpha\) be a nonzero complex number and assume that it is not a root of unity. Let \(R(X,Y)\) be a rational function complex coefficients and with expansion \(R(X,Y) = r_ 0 Y^ l + r_ 1(X) Y^{l-1} + \dots\). Consider the equation \[ \alpha^ x = R(x,y) \text{ in }x,y\in \mathbb{Z}.\tag{1} \] The author shows, that (1) has only finitely many solutions, except if \(R(X,Y)\) is of a special form. More exactly (Theorem 1), if (1) has solutions with arbitrarily large \(| x|\), then \(R(X,Y) = r_ 0(Y - h(X))^ l\) with \(l \neq 0\), \(h(X) \in \mathbb{Q}[X]\) and \(\alpha^ u \in \mathbb{Z}\) for some \(0 \neq u \in \mathbb{Z}\). Moreover (Theorem 2), if \(\alpha^ x = r_ 0(y - h(x))^ l\) has infinitely many solutions, then apart from finitely many exceptions they belong to finitely many one-parameter families. The proof of the above deep theorems uses results of \textit{M. Laurent} [Astérisque 147/148, 121-139 (1987; Zbl 0621.10014)] and \textit{H. P. Schlickewei} [Arch. Math. 29, 267-270 (1977; Zbl 0365.10026)].
rational function with complex coefficients
scientific article; zbMATH DE number 599849