Equations in one variable over function fields (Q2717764)

Let \(k\) be a field of characteristic \(0\), \(K=k(t)\) the field of rational functions in one variable over \(k\) and \(p_{1}>\cdots>p_{n}\geq 1\) positive integers with \(n\geq 2\). The authors first consider the equation NEWLINE\[NEWLINEa_{1}x^{p_{1}}+\cdots+a_{n}x^{p_{n}}=1, \tag{1}NEWLINE\]NEWLINE where \(a_{1},\ldots,a_{n}\) are given nonzero elements of \(K\) and the unknown \(x\) is a nonzero element in \(K\). Assume \(p_{i}-p_{i+1}>\lambda\) \ (\(1\leq i\leq n\)) where \(p_{n+1}=0\) and NEWLINE\[NEWLINE \lambda=(1/2)(n+1)!\bigl((n+1)!-1\bigr) \bigl((n+1)!-2\bigr). NEWLINE\]NEWLINE They prove that there are at most \(n\) solutions \(x\) in \(K\) to (1) which are pairwise non \(k\)-proportional. NEWLINENEWLINENEWLINENext they consider the equation NEWLINE\[NEWLINEa_{1}(m)\alpha_{1}^{m}+\cdots+ a_{n}(m)\alpha_{n}^{m}=0,\tag{2}NEWLINE\]NEWLINE where \(a_{1},\ldots,a_{n}\) are given nonzero polynomials in \(K[X]\), while \(\alpha_{1},\ldots,\alpha_{n}\) are given nonzero elements in \(K\) and the unknown \(m\) is in \(\mathbb Z\). They prove that if \(\alpha_{1},\ldots,\alpha_{n}\) are pairwise non \(k\)-proportional, then the number of solutions \(m\) in \(\mathbb Z\) to (2) is at most \(d_{1}+\cdots+d_{n}+n(n-1)/2\), where \(d_{i}\) is the degree of the polynomial \(a_{i}\). They also achieve the same conclusion under a weaker assumption on the \(\alpha_{i}\) than non \(k\)-proportionality. NEWLINENEWLINENEWLINEThey show, by means of examples, that their estimates are sharp. They also point out that for number fields, a uniform bound for the number of solutions to (2) has been achieved by \textit{W. M. Schmidt} [Acta Math. 182, No. 2, 243-282 (1999; Zbl 0974.11013].