  Consider a nontrivial solution to a semilinear elliptic system of first order
with smooth coefficients defined over an $n$-dimensional manifold. Assume the
operator has the strong unique continuation property. We show that the zero set
of the solution is contained in a countable union of smooth $(n-2)$-dimensional
submanifolds. Hence it is countably $(n-2)$-rectifiable and its Hausdorff
dimension is at most $n-2$. Moreover, it has locally finite $(n-2)$-dimensional
Hausdorff measure. We show by example that every real number between 0 and
$n-2$ actually occurs as the Hausdorff dimension (for a suitable choice of
operator). We also derive results for scalar elliptic equations of second
order.
