MathNet 2023-part-02 — 191 problems (EN–VI) Question 61

Determine all injective functions $f$ from $\mathbb{N}$ to $\mathbb{N}$ such that every polynomial $\sum_{i=0}^{n} a_{i} x^{i}$ has a real root if and only if the polynomial $\sum_{i=0}^{n} a_{i} x^{f(i)}$ has a real root.