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We use the Aron-Berner extension to prove that the set of extreme points ofthe unit ball of the space of integral polynomials over a real Banach space $X$is $\{\pm \phi^k: \phi \in X^*, \| \phi\|=1\}$. With this description we showthat, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in$Y$, then $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} X$ (the $k$-th symmetrictensor product of $X$ endowed with the injective symmetric tensor norm) is\emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{\epsilon_{k,s}} Y$. Thisresult marks up a difference with the behavior of non-symmetric tensors since,when $X$ is an $M$-ideal in $Y$, it is known that$\hat\bigotimes^k_{\epsilon_k} X$ (the $k$-th tensor product of $X$ endowedwith the injective tensor norm) is an $M$-ideal in$\hat\bigotimes^k_{\epsilon_k} Y$. Nevertheless, if $X$ is Asplund, we provethat every integral $k$-homogeneous polynomial in $X$ has a unique extension to$Y$ that preserves the integral norm. We explicitly describe this extension.  We also give necessary and sufficient conditions (related with the continuityof the Aron-Berner extension morphism) for a fixed $k$-homogeneous polynomial$P$ belonging to a maximal polynomial ideal $\Q(^kX)$ to have a unique normpreserving extension to $\Q(^kX^{**})$. To this end, we study the relationshipbetween the bidual of the symmetric tensor product of a Banach space and thesymmetric tensor product of its bidual and show (in the presence of the BAP)that both spaces have `the same local structure'. Other applications to themetric and isomorphic theory of symmetric tensor products and polynomial idealsare also given.

Geometry of integral polynomials, $M$-ideals and unique norm preserving extensions