On the equation \(F(x,y)=z^ m\) over function fields
Let K be an arbitrary finite extension of the one-dimensional rational function field k(t) over an algebraically closed field k of characteristic 0, and let S be a finite set of (additive) valuations v of K/k containing all the infinite valuations, with cardinality \(| S|\). Let \({\mathcal O}_ S\) denote the ring of S-integers of K. The (additive) height of an element a of K is defined to be \(H(a)=- \sum_{v}\min (0,v(a)),\) where v runs through the valuations of K/k with value group \({\mathbb{Z}}\). For \(f(x)\in K[x]\) \(v(f)\) is defined in an obvious way in terms of the values of its coefficients. We consider the equation (*): \(F(x,y)=cz^ m\), where \(F(X,Y)\) is a binary form of degree N with coefficients in K and \(c\neq 0\) is an element of K. Let L be the splitting field of \(f(X)=c^{-1}F(X,1)\) over K and denote by \(g(L)\) the genus of \(L/k.\) Improving and extending a previous result of \textit{W. M. Schmidt} [Queen's Pap. Pure Appl. Math. 54, 33-65 (1980; Zbl 0454.10010)], the author obtains the following theorem. Suppose that the form \(F(X,Y)\) has at least three distinct simple linear factors over L. Then all solutions x,y,z,m of the equation (*) with \(z\in {\mathcal O}_ S\), \(m\in {\mathbb{Z}}\), \(m>3\), and relatively prime x,y\(\in {\mathcal O}_ S\) satisfy \[ \max \{H(x),H(y),H(z)\}<1000 M^ 5, \] and if \(z\not\in k\) then \(m<M+N\), where \[ M=2N^ 3H(f)+2N((L:K)| S| +g(L)),\quad N\geq 3. \]
scientific article; zbMATH DE number 3991592
scientific article