Ideal structures in vector-valued polynomial spaces

This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, ${\mathcal{P}}_w{(}^{n}E,F)$, is an HB-subspace or an $M(1,C)$-ideal in the space of continuous $n$-homogeneous polynomials, $\mathcal{P}{(}^{n}E,F)$. We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from ${\mathcal{P}}_w{(}^{n}E,F)$ as an ideal in $\mathcal{P}{(}^{n}E,F)$ to the range space $F$ as an ideal in its bidual $F^{\ast \ast }$.