Hermitian forms with real coefficients

Question

Let $z$ be the $n$-tuple of complex variables $(z_1,\ldots,z_n)$ and define $H:{\bf C}^n\to{\bf R}$ by

$ H(z)=\sum_{i=1}^p |P_i(z)|^2 - \sum_{j=1}^q |Q_j(z)|^2, $

where $P_i,Q_j \in {\bf R} [z_1,\ldots,z_n]$ are homogeneous polynomials of the same degree, $d$.

For $\Omega=\{z\in {\bf C}^n: H(z)>0\}$, is it true that

$ \Omega \cap {\bf R}^n=\emptyset\quad \Longrightarrow\quad \Omega=\emptyset\;? $

Remarks: ${\bf R}$ real numbers, ${\bf C}$ complex numbers. All coefficients of the $P_i,Q_j$ are real. The degree $d$ is arbitrary.

Answer 1

For $d=1$ this is true because $H(z)$ is a bilinear form on $\mathbb{R}^n$ and nonpositive definite by assumption, so it is of the form $H(z) = -\lVert Az\rVert^2$ for some real matrix $A$, and therefore nonpositive definite on $\mathbb{C}^n$ as well. For a counterexample with $d=2$ it is enough to consider two variables and $p=1$ with $P_1(z_1, z_2) = 2z_1z_2$ and $Q_1(z_1, z_2) = z_1^2 + z_2^2$. Then $H(z_1, z_2) \leq 0$ for all $z_1, z_2\in\mathbb{R}$, but $H(1, i) = 4$.