Suppose that the real algebraic curve $\gamma$ in $\mathbb{R}^3$ is the intersection of the zero sets of the polynomials $p_1,...,p_k \in \mathbb{R}[x,y,z]$. Is the projection of $\gamma$ on a generic plane (isomorphic to $\mathbb{R}^2$) a real algebraic curve? I.e., is it the zero set of one polynomial $p \in \mathbb{R}[x,y]$?

It is obvious that it is not true that the projection of a real algebraic curve from $\mathbb{R}^2$ to $\mathbb{R}$ is again a real algebraic curve (if the initial curve is closed, then its projection is a line segment, which is not the zero set of a polynomial).

I would also like to ask:

Is the projection of a complex algebraic curve from $\mathbb{C}^3$ to $\mathbb{C}^2$ a complex algebraic curve?

I hope my questions are not trivial; I have searched in books, but I was unable to find an answer.

Thank you very much!