  We prove a characterization of a P$\star$MD, when $\star$ is a semistar
operation, in terms of polynomials (by using the classical characterization of
Pr\"{u}fer domains, in terms of polynomials given by R. Gilmer and J. Hoffman
\cite{Gilmer/Hoffmann:1974}, as a model), extending a result proved in the star
case by E. Houston, S.J. Malik and J. Mott \cite{Houston/Malik/Mott:1984}. We
also deal with the preservation of the P$\star$MD property by ``ascent'' and
``descent'' in case of field extensions. In this context, we generalize to the
P$\star$MD case some classical results concerning Pr\"{u}fer domains and
P$v$MDs. In particular, we reobtain as a particular case a result due to H.
Pr\"{u}fer \cite{Prufer} and W. Krull \cite{Krull:1936} (cf. also F. Lucius
\cite{Lucius:1998} and F. Halter-Koch \cite{Koch:2000}). Finally, we develop
several examples and applications when $\star$ is a (semi)star given explicitly
(e.g. we consider the case of the ``standard'' $v$--, $t$--, $b$--,
$w$--operations or the case of semistar operations associated to appropriate
families of overrings).
