expressing a quadratic map as a complex map

Question

Are there any known criterion when a real quadratic mapping $ Q:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n} $ can be expressed as a complex quadratic map $ Q:\mathbb{C}^n \rightarrow \mathbb{C}^n$?

Answer 1

Two polynomials agree on $\mathbb{R}^n$ iff they're the same polynomial, so if you have a polynomial $f(x_1,y_1,\ldots,x_n,y_n)$ then it's enough to consider the restriction

$$g(x_1,\ldots,x_n) := f(x_1,0, x_2,0,\ldots,x_n,0)$$

to the real axis, and then check if

$$g(x_1+y_1 i, \ldots, x_n + y_n i) = f(x_1,y_1,\ldots,x_n,y_n)$$