Schwartz-Zippel lemma ($\#\{x\in S^n:P(x)=0\}\le d|S{|}^{n-1}$)

For a nonzero multivariate polynomial of total degree $d$ over a field, and a finite subset $S$ of that field, at most $d|S{|}^{n-1}$ points of $S^n$ are zeros. For one variable this is the root bound for a polynomial. If $d\ge |S|$, the bound is already trivial. Otherwise, in the inductive proof write the polynomial as degree $k$ in its last variable, with a nonzero leading coefficient of degree at most $d-k$ in the other variables. The leading coefficient vanishes on at most $(d-k)|S{|}^{n-2}$ fibers; outside those fibers there are at most $k$ roots of a polynomial per fiber. Counting the bad fibers with the trivial $|S|$ bound gives the asserted inequality.