Weierstrass preparation of quadratic differentials

Special cases or variants of the following play an important role in the asymptotic analysis of ordinary differential equations with turning points.Theorem.Let a(t, x) be a smooth complex-valued function germ at the origin in C × R^mwhich is holomorphic in t. Suppose that a(t, 0) does not vanish identically. Then there is a smooth change of variable t = g(s, x), holomorphic in s, such that a(t, x)dt² =P(s, x)ds²where P is a monic polynomial in s.