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GCD of two functions a constant polynomial

Source: Canada Repêchage 2014/1

June 18, 2016
functionalgebrapolynomialnumber theorygreatest common divisor

Problem Statement

Let f:Z→Z+f : \mathbb{Z} \rightarrow \mathbb{Z}^+ be a function, and define h:Z×Z→Z+h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+ by h(x,y)=gcd⁡(f(x),f(y))h(x, y) = \gcd (f(x), f(y)). If h(x,y)h(x, y) is a two-variable polynomial in xx and yy, prove that it must be constant.
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