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!



Projections of real algebraic curves
As already mentioned, the answer is negative for varieties.


However, if you are interested in a more general setting when the answer is positive (i.e. a more general class of sets that *is* closed under projections), you can look at


* semialgebraic sets in the case of real closed fields (e.g. $\mathbb{R}$),
* constructible sets (boolean combinations of zero sets) in the case of algebraically closed fields (e.g. $\mathbb{C}$).


This is related to quantifier elimination. The theory of real closed fields (RCF) and the theory of algebraically closed fields (ACF) admit quantifier elimination (see e.g. *David Marker, Model Theory, §3.2, 3.3.*), which means that if you take a definable set in RCF or ACF, then its projection is also a definable set. And the definable sets are semialgebraic sets and constructible sets respectively.


