Let $R$ be a unitary commutative ring (or take a field if this is easier), denote by $\mathcal{S}_{R,d}$ the set of $P\in R[X]$, $P=X^d+z_{d-1}X^{d-1}+\cdots +z_0$, such that for all $i\in \{0,\ldots, d-1\}$, $P(z_i)=0$.
Have the sets $\mathcal{S}_{R,d}$ been studied, do they have a name, nice properties (e.g. finite for some cases)?
This is more of a curiosity than anything else. If $R$ is a field, for small degrees (i.e. $d=1, 2$ or $3$) a full characterization seems reachable.
