Zero sets of homogeneous polynomials containing infinite dimensional spaces

Resumo: In 1998, Plichko and Zagorodnyuk proved the following remarkable result: For any infinite dimensional complex linear space $X$, every $\mathbb{C}$-valued homogeneous polynomial on $X$ vanishes on an infinite dimensional subspace of $X$. In the modern language of lineability, the Plichko-Zagorodnyuk theorem asserts that the zero set of any homogeneous polynomial on an infinite dimensional complex space is lineable, meaning that it contains an infinite dimensional linear space. In this talk, we will 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.  We will also discuss applications to the real case. This is a joint work with Geraldo Botelho.