IR-TST 2023 Problem 11

Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0$ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$ ), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.

Proposed by Hesam Rajabzadeh