Question 47
Find all injective functions $f : \mathbb{Z}^{\ge 0} \rightarrow \mathbb{Z}^{\ge 0}$ such that for all $n \in \mathbb{N}$ and every $(n+1)$-tuple of real numbers like $(a_0, \dots, a_n)$ where not all of them are equal to zero, the 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.