Let \(P:E \to \mathbb K = \mathbb C\) be a polynomial with \(P(0) = 0\) where \(E\) is an infinite dimensional complex Banach space. It is known that \(P^{-1}(0)\) contains an infinite dimensional subspace. However it is an open problem if \(P^{-1}(0)\) contains a non-separable subspace if \(E\) is non-separable. The author answers this question here in the affirmative when \(E = \ell_\infty.\) In addition, the author shows that if we consider the real \(\ell_\infty,\) then a polynomial \(P:\ell_\infty \to \mathbb R\) that vanishes on a copy of \(c_0\) necessarily vanishes on a non-separable subspace of \(\ell_\infty.\) One interesting side consequence is that if \(c_0 \subset F \subset \ell_\infty\) is separable, then \(F\) is not the intersection of a countable family of zero sets \(P_j^{-1}(0)\) where each \(P_j:\ell_\infty \to \mathbb K\) is a polynomial. The paper also contains a discussion of so-called maximal subspaces relative to a polynomial: For an \(n\)-homogeneous polynomial \(P:E \to \mathbb K,\) the subspace \(F \subset E\) is said to be maximal relative to \(P\) if \(F \subset P^{-1}(0)\) and no larger subspace is contained in \(P^{-1}(0).\)
Reviewer’s note. For some background information, see the reviewer’s article with C. Boyd, R. A. Ryan and I. Zalduendo [Positivity 7, 285–295 (2003; Zbl 1044.46036)] and T. Banakh, A. Plichko and A. Zagorodnyuk [Colloq. Math. 100, 141–147 (2004; Zbl 1066.46040)].