Let $p\in \mathbb{R}[x_1, \ldots, x_n]$ be a real (non-constant) multilinear polynomial of degree $\deg p=d$, then its real zero-set $V_{\mathbb{R}}(p)$ is non-empty and has dimension $n-1$ by the sign-changing criterion (Marshall, Positive Polynomials and Sums of Squares, Theorem 12.7.1.). It's also known that $V_{\mathbb{R}}(p)$ has at at most $2^{d-1}$ connected components (Theorem 2.). Assuming $n>1$, does anyone know the answer to the following questions.
- Can the zero-set $V_{\mathbb{R}}(p)$ have a compact connected component? I'm pretty sure all connected components are unbounded if we additionally assume the polynomial is symmetric, but I have no idea for the general case.
- Can the zero-set $V_{\mathbb{R}}(p)$ contain connected components of dimension $<n-1$ that are not contained in a coordinate hyperplane? (Can it contain any at all?)
Any help would be greatly appreciated.