  Let $E$ be a Banach space and, for any positive integer $n$, let ${\cal
P}(^nE)$ denote the Banach space of continuous $n$-homogeneous polynomials on
$E$. Davie and Gamelin showed that the natural extension mapping from ${\cal
P}(^nE)$ to ${\cal P}(^nE^{\ast\ast})$ is an isometry into the latter space.
Here, we investigate when there is a natural isomorphism between ${\cal
P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$. Among other things, we show
that if $E$ satisfies: \break (a) no spreading model built on a normalised
weakly null sequence has a lower $q$-estimate for any $q < \infty,$ (b)
$E^{\ast}$ has RNP, and (c) $E^{\ast}$ has the approximation property, then
${\cal P}(^nE)$ has RNP for every $n$. Moreover, if $E$ satisfies (a) and is
such that $E^{\ast\ast}$ has both the RNP and the approximation property, then
${\cal P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$ are isomorphic for
every $n$. We also exhibit a quasi-reflexive Banach space $E$ for which ${\cal
P}(^nE)^{\ast\ast}$ and ${\cal P}(^nE^{\ast\ast})$ are isomorphic for every
$n$.
  Related work has been done recently by (i) M. Gonzalez, (ii) M. Valdivia, and
(iii) J. Jaramillo, A. Prieto, and I. Zalduendo.
