Magic sets for polynomials of degree \(n\) (Q2226469)

Denote by \(\mathcal{P}_n\) the family of all real, non-constant polynomials with degree at most \(n,\) and by \(\mathcal{Q}_n\) the family of all complex, non-constant polynomials with degree at most \(n.\) A set \(S \subseteq \mathbb{R}\) is called a set of range uniqueness (SRU) for \(\mathcal{P}_n\) \((\mathcal{Q}_n)\) if for all \(f, g \in \mathcal{P}_n\) (\(f, g \in \mathcal{Q}_n\)) the following is true: \(f[S]=g[S] \Rightarrow f=g.\) A set \(S \subseteq \mathbb{R}\) is called a magic set for \(\mathcal{P}_n\) (\(\mathcal{Q}_n\) ) if for all \(f, g \in \mathcal{P}_n\) (\(f, g \in \mathcal{Q}_n\)): \(f[S] \subseteq g[S] \Rightarrow f=g.\) In this paper, the authors show that there exist no SRUs, and therefore also no magic sets, of size at most \(2n\) for \(\mathcal{P}_n\) and \(\mathcal{Q}_n.\) They prove that for every \(s\geq 2n+1\) there exists a magic set of size \(s\) for the families \(\mathcal{P}_n\) and \(\mathcal{Q}_n.\) The authors also construct examples of SRUs for \(\mathcal{P}_2\) and \(\mathcal{P}_3\) that are not magic.