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 ^m which 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)dt ² =P(s, x)ds ² where P is a monic polynomial in s.