  We consider sparse polynomials in $N$ variables over a finite field, and ask
whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of
the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$
with $f (\mathbf{c}) \neq 0$, then there is such a $\mathbf{c}$ in every sphere
inside $S^N$, where the radius of the sphere is bounded by a multiple of the
logarithm of the number of monomials that appear in $f$. A similar result holds
for the solutions of the equations $f_1 = \cdots = f_r = 0$ inside $S^N$.
