Let $f(X)$ be a polynomial with rational coefficients, $S$ be an infinite subset of the rational numbers and consider the image set $f(S)$. If $g(X)$ is a polynomial such that $f(S)=g(S)$ we say that $g$ \emph{parametrizes} the set $f(S)$. Besides the obvious solution $g=f$ we may want to impose some conditions on the polynomial $g$; for example, if $f(S)\subset\Z$ we wonder if there exists a polynomial with integer coefficients which parametrizes the set $f(S)$. Moreover, if the image set $f(S)$ is parametrized by a polynomial $g$, there comes the question whether there are any relations between the two polynomials $f$ and $g$. For example, if $h$ is a linear polynomial and if we set $g=f\circ h$, the polynomial $g$ obviously parametrizes the set $f(\Q)$. Conversely, if we have $f(\Q)=g(\Q)$ (or even $f(\Z)=g(\Z)$) then by Hilbert's irreducibility theorem there exists a linear polynomial $h$ such that $g=f\circ h$. Therefore, given a polynomial $g$ which parametrizes a set $f(S)$, for an infinite subset $S$ of the rational numbers, we wonder if there exists a polynomial $h$ such that $f=g\circ h$. Some theorems by Kubota give a positive answer under certain conditions. The aim of this thesis is the study of some aspects of these two problems related to the parametrization of image sets of polynomials.
