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We consider sparse polynomials in $N$ variables over a finite field, and askwhether they vanish on a set $S^N$, where $S$ is a set of nonzero elements ofthe 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 sphereinside $S^N$, where the radius of the sphere is bounded by a multiple of thelogarithm of the number of monomials that appear in $f$. A similar result holdsfor the solutions of the equations $f_1 = \cdots = f_r = 0$ inside $S^N$.

Zero testing and equation solving for sparse polynomials on rectangular domains