  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.
