[Math] Facts from algebraic geometry that are useful to non-algebraic geometers

Question

A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic information about Galois theory and algebraic geometry, which, according to him, are important even for non-algebraists.

What are some useful facts from algebraic geometry that are useful for non-algebraic geometers? Ideally, the statements at least should be accessible without knowing much algebraic geometry.

Edit: Please do not post results that are only relevant to people who already know massive amounts of algebraic geometry anyway. In particular: Be very cautious about posting statements whose only applications are in number theory.

Example: Here is a basic statement that I have seen applied outside algebraic geometry, if not necessarily outside of algebra:

Let $U \subset \mathbb{C}^n$. If there is some nonzero polynomial satisfied by every point of $\mathbb{C}^n \smallsetminus U$, then $U$ is dense in $\mathbb{C}^n$ (with the usual topology), and in fact contains a dense open subset of $\mathbb{C}^n$.

[Sketch of proof: Given any point $p \in \mathbb{C}^n$, find a complex line $L$ passing through $p$ that intersects $U$. Then $L \cap (\mathbb{C}^n \smallsetminus U)$ is algebraic, hence contains only finitely many points of $L$, and so $p$ is a limit point of $U$.]

Answer 1

I would vote for Chevalley's theorem as the most basic fact in algebraic geometry:

The image of a constructible map is constructible.

More down to earth, its most basic case (which, I think, already captures the essential content), is the following: the image of a polynomial map $\mathbb{C}^n \to \mathbb{C}^m$, $z_1, \dots, z_n \mapsto f_1(\underline{z}), \dots, f_m(\underline{z})$ can always be described by a set of polynomial equations $g_1= \dots = g_k = 0$, combined with a set of polynomial ''unequations'' (*) $h_1 \neq 0, \dots, h_l \neq 0$.

David's post is a special case (if $m > n$, then the image can't be dense, hence $k > 0$). Tarski-Seidenberg is basically a version of Chevalley's theorem in ''semialgebraic real geometry''. More generally, I would argue it is the reason why engineers buy Cox, Little, O'Shea ("Using algebraic geometry"): in the right coordinates, you can parametrize the possible configurations of a robotic arm by polynomials. Then Chevalley says the possible configuration can also be described by equations.

(*) Really it seems that "inequalities" would be the right word her...might be a little late to change terminology though...