Max–min of polynomials and exponential diophantine equations

Abstract

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, max x ∈ Z ( T ) min y ∈ Z | F ( x , y ) | = o ( T 1 / 2 ) as T → ∞ if and only if there is a positive integer B such that max x ∈ Z min y ∈ 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 ) b G ( 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 ) ) b G ( x , h ( x ) ) ≡ 0 , and h ( α ) ∈ Z .