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Let $f,g\in C^\infty(\mathbb R^n;\mathbb R)$ be two Morse functions having both a critical point at $0$. Is it always possible to find local coordinates around $0$ such that both $f $ and $g$ become quadratic in the new coordinates?

After the comment of Matt, I realized that i forgot an important assumption: $0$ is a critical point of index $0$ of $f$.

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    @Matt. many thanks for your comment, I edited the question. Now, at least on the level of quadratic forms, simultaneous diagonalization is always possible.2012-08-09

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Take $f=g+g^2$ where $g$ is your favorite Morse function. This relation is not affected by coordinate changes. If $g$ is quadratic then $f$ isn't.

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    Many thanks for this perfect answer @Leonid Kovalev.2012-08-12