Zero sets of solutions to semilinear elliptic systems of first order

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