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Roots, bounding and other delusions

Source: INMO 2021 Problem 6

March 7, 2021
PolynomialsfunctionsReal RootsalgebraINMO

Problem Statement

Let R[x]\mathbb{R}[x] be the set of all polynomials with real coefficients. Find all functions f:R[x]→R[x]f: \mathbb{R}[x] \rightarrow \mathbb{R}[x] satisfying the following conditions:
[*] ff maps the zero polynomial to itself, [*] for any non-zero polynomial P∈R[x]P \in \mathbb{R}[x], deg f(P)≤1+deg P\text{deg} \, f(P) \le 1+ \text{deg} \, P, and [*] for any two polynomials P,Q∈R[x]P, Q \in \mathbb{R}[x], the polynomials P−f(Q)P-f(Q) and Q−f(P)Q-f(P) have the same set of real roots.
Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha
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