The stationary phase method for real analytic geometry

We prove that the existence of isolated solutions of systems of equations of analytical functions on compact real domains in RpRp, is equivalent to the convergence of the phase of a suitable complex valued integral I(h)I(h) for h→∞h→∞. As an application, we then use this result to prove that the problem of establishing the irrationality of the value of an analytic function F(x)F(x) at a point x0x0 can be rephrased in terms of a similar phase convergence.