# An Exercise on Inverse Function Theorem

Consider the mapping $f:R^2\rightarrow R^2$ given componentwise by:

$f_1(x,y)=x+a_1x^2+2b_1xy+c_1y^2\\ f_2(x,y)=y+a_2x^2+2b_2xy+c_2y^2$

Determine a neighbourhood of $(0,0)$ as large as possible on which $f$ is invertible and bijective.

Inverse function theorem assures the existence of such neighbourhood. However, how do I choose the largest neighbourhood such that this holds?

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Consider the behaviour of the partial derivatives... –  Edward Hughes May 3 '12 at 23:15

Let $R$ be the largest number such that $F=(f_1,f_2)$ is injective in $\{x\colon |x|<R\}$. I don't know how to find $R$ precisely in terms of $a_i,b_i,c_i$ (and suspect there is no nice formula for it), but here are some estimates.
1) You can get a lower bound for $R$ from the following observation: if $F(\vec x)=\vec x+G(\vec x)$ where $G$ satisfies $|G(\vec x)-G(\vec y)|<|\vec x-\vec y|$ for all $\vec x,\vec y$ in its domain, then $F$ is injective. In particular, this applies when the Jacobian matrix of $G$ has norm strictly less than $1$. The matrix is $$DG(x,y)=2\begin{pmatrix}a_1x+b_1y & b_1x+c_1y \\ a_2x+b_2y & b_2x+c_2y \end{pmatrix}$$ However, the operator norm of a matrix, even of size $2\times 2$, can be messy. If we use the somewhat sloppy estimate by the sum of $|x|<R$, $|y|<R$, and bound the operator norm by the sum of the norms of rows, we'll get $$\|DG\|\le 2R\left(\sqrt{(|a_1|+|b_1|)^2+(|b_1|+|c_1|)^2}+\sqrt{(|a_2|+|b_2|)^2+(|b_2|+|c_2|)^2}\right)$$ So, the map is assured to be invertible when $$R<\frac{1}{2}\left(\sqrt{(|a_1|+|b_1|)^2+(|b_1|+|c_1|)^2}+\sqrt{(|a_2|+|b_2|)^2+(|b_2|+|c_2|)^2}\right)^{-1}$$
2) Let's also get an upper bound on $R$. In general, the vanishing of Jacobian determinant does not imply the failure of injectivity. However, our map is a polynomial of 2nd degree. Suppose we expand it at a point where the Jacobian determinant vanishes. Then the linear part has a kernel, and the restriction of the map onto the kernel is of the form $(\alpha t^2,\beta t^2)$. Hence, the map is not injective in any neighborhood of such point. The set $\det DF=0$ is a quadric surface, so it is a feasible task to find the nearest point to $(0,0)$ on this surface. Again, this does not look like a pleasant task... I have to say, if this is a textbook/homework exercise, it appears to be poorly thought out.