On decompositions of quadrinomials and related Diophantine equations

Maciej Gawron

paper

On decompositions of quadrinomials and related Diophantine equations

arXiv:1512.02817

Abstract

Let $A, B, C, D$ be rational numbers such that $A B C \neq = 0$ , and let $n_{1} > n_{2} > n_{3} > 0$ be positive integers. We solve the equation $A x^{n_{1}} + B x^{n_{2}} + C x^{n_{3}} + D = f (g (x)),$ in $f, g \in Q [x]$ . In sequel we use Bilu-Tichy method to prove finitness of integral solutions of the equations $A x^{n_{1}} + B x^{n_{2}} + C x^{n_{3}} + D = E y^{m_{1}} + F y^{m_{2}} + G y^{m_{3}} + H,$ where $A, B, C, D, E, F, G, H$ are rational numbers $A B C E F G \neq = 0$ and $n_{1} > n_{2} > n_{3} > 0$ , $m_{1} > m_{2} > m_{3} > 0$ , $g cd (n_{1}, n_{2}, n_{3}) = g cd (m_{1}, m_{2}, m_{3}) = 1$ and $n_{1}, m_{1} \geq 9$ . And the equation $A_{1} x^{n_{1}} + A_{2} x^{n_{2}} + \dots + A_{l} x^{n_{l}} + A_{l + 1} = E y^{m_{1}} + F y^{m_{2}} + G y^{m_{3}},$ where $l \geq 4$ is fixed integer, $A_{1}, \dots, A_{l + 1}, E, F, G$ are non-zero rational numbers, except for possibly $A_{l + 1}$ , $n_{1} > n_{2} > \dots > n_{l} > 0$ , $m_{1} > m_{2} > m_{3} > 0$ are positive integers such that $g cd (n_{1}, n_{2}, \dots n_{l}) = g cd (m_{1}, m_{2}, m_{3}) = 1$ , and $n_{1} \geq 4$ , $m_{1} \geq 2 l (l - 1)$ .

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