On common values of lacunary polynomials at integer points

For fixed ` ≥ 2, fixed positive integers m1 > m2 with gcd(m1,m2) = 1 and n1 > n2 > · · · > n` with gcd(n1, . . . , n`) = 1, and fixed rationals a1, a2, . . . , a`+1, b1, b2 which are all nonzero except for possibly a`+1, we show the finiteness of integral solutions x, y of the equation a1x n1 + · · ·+ a`x` + a`+1 = b1y + b2y , when n1 ≥ 3, m1 ≥ 2`(` − 1), and (n1, n2) 6= (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.