On solutions of real analytic equations

Analyticity of ${\c C}^{\infty }$ solutions $y_i =f_i(x), 1\le i\le m$, of systems of real analytic equations $p_j(x,y)= 0, 1\le j\le l$, is studied. Sufficient conditions for ${\c C}^{\infty }$ and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by $p_1,p_2,\ldots ,p_l$. In a special case when the $p_i$'s are independent of $x$, we prove that if a ${\c C}^{\infty }$ solution $h$ satisfies the condition $\det \left( \frac {\partial p_i}{py_j}\right )(h(x)) \not \equiv 0$, then $h$ is necessarily real analytic.