Non-equivalent germs remain non-equivalent when adding quadratic forms in new variables

Let two smooth real-valued functions f and g of several real variables have a totally degenerate critical point at the origin and assume their germs to be not equivalent, i.e. they do not locally coincide after any smooth coordinate transformation. Then the germs of f + q and g + q, where q is a quadratic form in new variables, are also not equivalent. It is often overlooked that this fact is non-trivial and a proof cannot be found in the literature, except for a rough sketch by Thom (Structural Stability and Morphogenesis, W.A. Benjamin, Reading, MA, 1975) which uses a result from an extensive theory by Tougeron (Ann. Inst. Fourier 18 (1) (1968) 177). The persistence of inequivalence under the addition of quadratic forms in new variables is essential for the classification of critical points in catastrophe theory. Our proof does not require deeper knowledge in algebra and is suited for advanced undergraduate text book presentation. (C) 2003 Elsevier Science Ltd. All rights reserved.