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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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    This probably boils down to the question of whether or not you can simultaneously diagonalize two particular matrices. You can probably cook up an example where you can't do this, or prove that the conditions of being Morse functions always guarantees simultaneous diagonalization (unlikely).2012-08-09
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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