Zero sets of homogeneous polynomials containing infinite dimensional spaces

TL;DR

This paper investigates conditions under which the zero set of a homogeneous polynomial in an infinite dimensional space extends from finite to infinite dimensions, advancing classical results especially in the complex case.

Contribution

It establishes new conditions linking finite and infinite dimensional zero sets of homogeneous polynomials, extending classical theorems in complex spaces.

Findings

Conditions identified for zero sets to extend to infinite dimensions

Extension of classical results by Plichko and Zagorodnyuk in complex spaces

Applications provided for the real case

Abstract

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.