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On the Diophantine equation \(f(x)f(y)=f(z)^n\) involving Laurent polynomials. II. (English) Zbl 1450.11028

In the paper the authors are interested in finding rational parametric solutions of the Diophantine equations of the form \((*)\; f(x)f(y)=f(z)\) and \((**)\; f(x)f(y)=f(z)^2\), where \(f\) is a Laurent polynomial of a specific form. For example, the authors prove that for \[ f(x)=x^{k}+ax^{k-1}+\frac{b}{x}, \] where \(a, b\in\mathbb{Q}\setminus\{0\}\) and \(k\geq 2\), the Diophantine equation \((*)\) has a rational parametric solution. If \(f(x)=x^2+a/x+b/x^2\), where \(a, b\in\mathbb Z\) and \(a\nmid 2b\), then the Diophantine equation \((*)\) has infinitely many rational solutions.
Similarly, if \[ f(x)=x^2+ax+b+\frac{a^{3}}{27x}, \] where \(a, b\in\mathbb{Q}\setminus\{0\}\), then the Diophantine equation \((**)\) has a rational parametric solution.
Part I cf. [Colloq. Math. 151, No. 1, 111–122 (2018; Zbl 1431.11050)].

MSC:

11D72 Diophantine equations in many variables
11D25 Cubic and quartic Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11G05 Elliptic curves over global fields

Citations:

Zbl 1431.11050
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References:

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