On decompositions of quadrinomials and related Diophantine equations

الأصل · EN

Let A,B,C,D be rational numbers such that ABC ≠ 0, and let n₁>n₂>n₃>0 be positive integers. We solve the equation Axⁿ¹+Bxⁿ²+Cxⁿ³+D = f(g(x)), in f,g ∈ Q[x]. In sequel we use Bilu-Tichy method to prove finitness of integral solutions of the equations Axⁿ¹+Bxⁿ²+Cxⁿ³+D = Eyᵐ¹+Fyᵐ²+Gyᵐ³+H, where A,B,C,D,E,F,G,H are rational numbers ABCEFG ≠ 0 and n₁>n₂>n₃>0, m₁>m₂>m₃>0, (n₁,n₂,n₃) = (m₁,m₂,m₃)=1 and n₁,m₁ ≥ 9. And the equation A₁xⁿ¹+A₂xⁿ²++Aₗ xⁿˡ + Aₗ₊₁ = Eyᵐ¹+Fyᵐ²+Gyᵐ³, where l ≥ 4 is fixed integer, A₁,,Aₗ₊₁,E,F,G are non-zero rational numbers, except for possibly Aₗ₊₁, n₁>n₂> > nₗ>0, m₁>m₂>m₃>0 are positive integers such that (n₁,n₂, nₗ) = (m₁,m₂,m₃)=1, and n₁ ≥ 4, m₁ ≥ 2l(l-1).