An upper bound for the size of integral solutions to \(Y^ m= f(X)\)
Let \(\mathbb{K}\) be an algebraic number field of degree \(\delta\) and discriminant \(D\). The author studies the solutions of the equation \((*)\) \(y^m= f(x)\), \(f(x)\in \mathbb{K} [x]\), \(\deg f=n\), in integers of \(\mathbb{K}\). Let \(e_1, e_2, \dots\) be the multiplicities of the roots of \(f(x) =0\) and put \(m_i= m/ (e_i, m)\), \(i=1, 2, \dots\;\). \textit{W. J. LeVeque} [On the equation \(y^m= f(x)\), Acta Arith. 9, 209--219 (1964; Zbl 0127.27201)]\ has shown that \((*)\) can have infinitely many integral solutions in \(\mathbb{K}\) only if \((m_1, m_2, m_3, \dots)\) is a permutation of either \((2, 2, 1, 1, \dots)\) or of \((t, 1, 1, \dots)\) with \(t\geq 1\). In the opposite case, one has essentially two subcases. Either (a): \(m_1\leq 2\) for each \(i\) and \(m_1= m_2= m_3= 2\), or (b): \(m_1\geq 3\) and \(m_2\geq 2\). These conditions are necessary and sufficient for the curve defined by \((*)\) to have positive genus. Let \(F(x, y)= y^m- f(x)\); define as usual \(H_\mathbb{K} (x)\), \(H_\mathbb{K} (y)\) and \(H_\mathbb{K} (F)\), the heights of the elements \(x\), \(y\) and of the polynomial \(F\), respectively, relative to the field \(\mathbb{K}\). Then, the author proves that \(\max (H_\mathbb{K} (x), H_\mathbb{K} (y))\) is bounded by something which is of the size of \(V_1 (\log V_1 )^{6n^2 \delta}\) in case (a) and of the size of \(V_3(\log V_3)^{m^6 n^2 \delta}\) in case (b); here \(V_1= D^{6n^2} H_\mathbb{K} (F)^{30 n^2}\) and \(V_3= (|D|\cdot H_\mathbb{K} (F))^{m^8 n^2/3}\). Of course, in the paper, there are precise formulations of the corresponding results. The proof, according to the author ``\dots follows the classical argument of \textit{C. L. Siegel} [J. Lond. Math. Soc. 1, 66--68 (1926; JFM 52.0149.02)], using upper bounds for linear forms in logarithms''.
scientific article; zbMATH DE number 798701
scientific article