Max–min of polynomials and exponential diophantine equations (II)

Abstract

Based on the estimate of max–min values of polynomials over integers, we study exponential diophantine equations with parameters. We proved the case for exponential equations with two terms. In this paper, we extend our previous results to general exponential equations. In particular, we have the following: Let F(x,y)=∑k=0mfk(x,y)bGk(x,y), where fk(x,y) and Gk(x,y) are in Z[x,y], b is a positive integer, and Gi(x,y)−Gj(x,y)∈(Z[x,y]−Z[x])∪Z for each 1≤i≠j≤m. Then for every integer α there is an integer c such that F(α,c)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that F(x,h(x))≡0 and h(α)∈Z. This result can be extended to higher orders of exponential diophantine equations.