Problem

Source: Iran 3rd round 2014-Algebra exam-P5

Tags: algebra, polynomial, algebra proposed



We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is good if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\] Proposed by Mohammad Ahmadi