Theoretical Computer Science

# Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of SchwartzZippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz.

Are there any other proofs which use substantially different ideas?

## Solution

Here's another idea I had for a geometric proof. It uses projective geometry in an essential way.

Let $c \in \mathbb F^m$ be an affine point outside the hypersurface $S$. Project the hypersurface onto the hyperplane at infinity using $c$ as center; that is, map every $x \in S$ onto $p(x)$, the intersection of the unique line through $c$ and $x$ with the hyperplane at infinity. The preimages under $p$ of a point at infinity all lie on the same line, and therefore (again reducing the problem to dimension 1) there are most $d$ of them. The hyperplane at infinity has cardinality $|\mathbb F^{m-1}|$, so we get the familiar upper bound $|S| \leq d\ |\mathbb F^{m-1}|$.