On decompositions of quadrinomials and related Diophantine equations

TL;DR

This paper investigates the decomposition of quadrinomials and related Diophantine equations, proving finiteness of solutions and classifying possible decompositions using advanced algebraic methods.

Contribution

It provides new results on polynomial decompositions and finiteness of solutions for complex Diophantine equations involving multiple exponents.

Findings

Finite integral solutions for specific polynomial equations.

Complete classification of polynomial decompositions.

Application of Bilu-Tichy method to Diophantine equations.

Abstract

Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let $n_1>n_2>n_3>0$ be positive integers. We solve the equation $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In sequel we use Bilu-Tichy method to prove finitness of integral solutions of the equations $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}+H, $$ where $A,B,C,D,E,F,G,H$ are rational numbers $ABCEFG \neq 0$ and $n_1>n_2>n_3>0$, $m_1>m_2>m_3>0$, $\gcd(n_1,n_2,n_3) = \gcd(m_1,m_2,m_3)=1$ and $n_1,m_1 \geq 9$. And the equation $$ A_1x^{n_1}+A_2x^{n_2}+\ldots+A_l x^{n_l} + A_{l+1} = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}, $$ where $l \geq 4$ is fixed integer, $A_1,\ldots,A_{l+1},E,F,G$ are non-zero rational numbers, except for possibly $A_{l+1}$, $n_1>n_2>\ldots > n_l>0$, $m_1>m_2>m_3>0$ are positive integers such that $\gcd(n_1,n_2, \ldots n_l) = \gcd(m_1,m_2,m_3)=1$, and $n_1 \geq 4$, $m_1 \geq 2l(l-1)$.