Zero sets of homogeneous polynomials containing infinite dimensional spaces

Let X be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial P on X so that, if W is any finite dimensional subspace of X on which P vanishes, then P vanishes on an infinite dimensional subspace of X containing W. In the complex case, this is a step beyond the classical result due to Plichko and Zagorodnyuk. Applications to the real case are also provided.