For fixed ℓ≧ 2, fixed positive integers m₁> m₂ with gcd(m₁, m₂)=1 and n₁>n₂> ... >n_ℓ with gcd(n₁, ..., n_ℓ)=1, and fixed rationals a₁, a₂, ..., a_ℓ+1, b₁, b₂ which are all nonzero except for possibly a_ℓ+1, we show the finiteness of integral solutions x, y of the equation a₁x^n₁+...+a_ℓx^n_ℓ+a_ℓ+1=b₁y^m₁+b₂y^m₂, when n₁≧ 3, m₁≧ 2ℓ(ℓ-1), and (n₁, n₂)≠ (m₁, m₂). In relation to that, we show the finiteness of integral solutions of equations of type f(x)=g(y), where f, g∈ Q[x] are of distinct degrees ≧ 3, and are such that they have distinct critical points and distinct critical values.

The author is thankful for the support of the Austrian Science Fund (FWF) via projects W1230-N13, FWF-P24302 and F5510.