On decompositions of quadrinomials and related Diophantine equations

Let $A,B,C,D$ be rational numbers such that $ABC\mathrm{\ne }0$, 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 $f,g\in \mathbb{Q}[x]$. 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 $ABCEFG\mathrm{\ne }0$ 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 $n_1,m_1\ge 9$. 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$l \geq 4$isfixedinteger,$A_1,\ldots,A_{l+1},E,F,G$arenon-zerorationalnumbers,exceptforpossibly$A_{l+1}$,$n_1>n_2>\ldots > n_l>0$,$m_1>m_2>m_3>0$ are positive integers such that $\gcd (n_1,n_2,\dots n_l)=\gcd (m_1,m_2,m_3)=1$, and $n_1\ge 4$, $m_1 \geq 2l(l-1)$.