  The main purpose of this paper is solve polynomial equations that are
satisfied by (generalized) polynomials. More exactly, we deal with the
following problem: let $\mathbb{F}$ be a field with
$\mathrm{char}(\mathbb{F})=0$ and $P\in \mathbb{F}[x]$ and $Q\in \mathbb{C}[x]$
be polynomials. Our aim is to prove characterization theorems for generalized
polynomials $f\colon \mathbb{F}\to \mathbb{C}$ of degree two that also fulfill
equation \[ f(P(x))= Q(f(x)) \] for each $x\in \mathbb{F}$. As it turns out,
the difficulty of such problems heavily depends on that we consider the above
equation for generalized polynomials or for (normal) polynomials. Therefore,
firstly we study the connection between these two notions.
