If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,y)=0$ has $d$ real solutions (corresponding to the $d$ complex zeros of $f$).
Motivating question: if you are given $u(x,y),v(x,y)$ how could you recognize whether they came from a complexification $f(x+iy)$?
The answer is to check the Cauchy Riemann equations.
Actual (and more difficult) question: Suppose you are given new generators $g(x,y),h(x,y)$ for the ideal $\langle u(x,y),v(x,y) \rangle$ (where $u$ and $v$ are the real and imaginary parts of the complexification of a univariate polynomial). Although the solutions $g=h=0$ are the same as $u=v=0$, you can check that $g(x,y),h(x,y)$ (generically) do not satisfy the Cauchy-Riemann equations. Now how can one tell when $g(x,y),h(x,y)$ came about in this fashion? and thus be able to tell immediately that they have real common solutions?
