Before revision:  With a simple modification, a version of Newton's iteration regains its local quadratic convergence to nonisolated zeros of smooth mappings assuming the solutions are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined  .
After revision:  With a simple modification, a version of Newton's iteration regains its local quadratic convergence to nonisolated zeros of smooth mappings assuming the solutions are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined while Jacobians can be rank-deficient .