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The following is an old qualifying exam question that has stumped me:

Let $f,g\in C^\infty(\mathbb{R}^3)$ be real-valued functions such that for some $x_0\in \mathbb{R}^3,$ we have $f(x_0) = g(x_0) = 0$ with $df(x_0)$ and $dg(x_0)$ linearly independent. Let $S_f$ and $S_g$ be the zero sets of $f$ and $g$ respectively. Show that if $h\in C^\infty(\mathbb{R}^3)$ vanishes on $S_f\cup S_g,$ then there exists a neighborhood $U\ni x_0$ such that in $U,$ we have $h(x) = f(x)g(x)H(x),$ for some $H\in C^\infty(U)$.

My initial thought was that the implicit function theorem might be a good place to start, but I have not been able to make any progress on it. Any thoughts are much appreciated.

Fabien
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cathy
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    How do you show this if $f(x) = x_1,$ while $g(x) = x_2?$ (so, the coordinate functions)? – Igor Rivin Dec 31 '13 at 01:42
  • I don't understand your question above. – cathy Dec 31 '13 at 02:14
  • Which part do you not understand? – Igor Rivin Dec 31 '13 at 02:15
  • What are $x_1$ and $x_2$? – cathy Dec 31 '13 at 02:16
  • They are what I would ordinarily call $x$ and $y,$ except you had already used the letter $x.$ As I say, the coordinate functions. – Igor Rivin Dec 31 '13 at 02:18
  • Oh, okay. In that case, I still do not know - do you mean consider $h(x)$ as a function of $f(x),$ $g(x),$ and some third variable? – cathy Dec 31 '13 at 02:22
  • Yes, find a local coordinate system with $u_1=f(x)$, $u_2=g(x)$ and $u_3$ perpendicular to both. Then $h(x)=\tilde h(u)$ and you can explore the Taylor expansion of $\tilde h$. – Lutz Lehmann Dec 31 '13 at 08:04
  • How does it make sense to say that $u_3$ is perpendicular to both $f(x)$ and $g(x)$ when the latter are elements of $R$? I must be missing something about changing to a local coordinate system... – cathy Dec 31 '13 at 15:29

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Here's a warm-up question for you. Suppose $f\colon\Bbb R^2\to\Bbb R$ is smooth and vanishes on the $x$- and $y$-axes. How would we prove that $f(x,y) = xy g(x,y)$ for some smooth function $g$?

The important tool in multivariable analysis, sometimes called the $C^\infty$ trick, is that if $f\colon \Bbb R^n\to\Bbb R$ is smooth and vanishes at the origin, then we can write $f(x) = \sum\limits_{i=1}^n x_ig_i(x)$ for some smooth functions $g_i$. (A hint on how to prove this: Write $f(x)-f(0)=\int_0^1 \frac d{dt} f(tx)\,dt$.)

So you should be able to apply this (twice) to answer the question I posed in the first paragraph.

To get to this situation in your problem, you apply the inverse function theorem to obtain new coordinates $z_1,z_2,z_3$ on a neighborhood of the $x_0\in\Bbb R^3$ with $z=0 \iff x=x_0$ and $z_1=f(x)$, $z_2=g(x)$.

Ted Shifrin
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