Question 35
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients, and let $\operatorname{deg}P$ denote the degree of a nonzero polynomial $P$. Find all functions $f:\mathbb{R}[x]\to\mathbb{R}[x]$ satisfying the following conditions:
- $f$ maps the zero polynomial to itself.
- For every nonzero polynomial $P\in\mathbb{R}[x]$, $\operatorname{deg}f(P)\leq 1+\operatorname{deg}P$.
- For any polynomials $P,Q\in\mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.
- $f$ maps the zero polynomial to itself.
- For every nonzero polynomial $P\in\mathbb{R}[x]$, $\operatorname{deg}f(P)\leq 1+\operatorname{deg}P$.
- For any polynomials $P,Q\in\mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.