Max–min of polynomials and exponential diophantine equations

In this paper we study the maximum–minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|⩽B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.