\documentclass[reqno,12pt]{amsart}
\usepackage{amscd,amssymb}
\DeclareMathOperator{\image}{Im}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\bR}{{\mathbb R}}

\title{Math 23a Problem Set 8}
\dedicatory{due October 13th}

\begin{document}
1.Let $f(x,y)=x^3 +3x^2-12x +y^2-6y+12$. For any  $c\in \bR$ we denote
by $C_c \subset R^2$ the
level curve $C_c =\{ (x,y)\in \bR ^2 |f(x,y)=c\}$ .

 a)Show that the point $(2,2)$ is a smooth point of the curve  $C_0$.

b)Show there exists $\epsilon >0$ and differentiable  functions $f(x),g(y)$
defined on the interval $(2-\epsilon ,2+\epsilon)$ such that the curve
$C_0$ is equal to the graph of the function $y=f(x)$ and to also  the graph
of the function $ x=g(y)$ in a sufficiently small open  neighborhood of
the point   $(2,2)$.

c)Find the tangent line and the tangent space to  $C_0$ at $(2,2)$

d)Find all  $c\in \bR$ such that  the level curve $f(x,y)=c$ is not
smooth. 

e)For each  $c\in \bR$ such that  the level curve $f(x,y)=c$ is not
smooth find all the singular [not smooth] points on  $C_c$

f) Let $p(x),q(y)$ be polinomial of degree $m$ and $n$
correspondingly. For any  $c\in \bR$ consider the curve 
  $C_c =\{ (x,y)\in \bR ^2 |p(x)-q(y))=c\}$. Prove that there not more
then $mn$ values of $c$ for which the curve  $C_c$ is not smooth.

2. a)Let  $A=\begin{bmatrix}a&0\\0&b\end{bmatrix}$. Find for which $a,b \in
\bR$ there exists an open neighborhood $U\in M_2$ of the matrix $A^2$ and
a continuously differentiable map $g:U  \rightarrow M_2$ such that 
$g(X)^2=X \forall X\in U$.

b)EXTRA CREDIT. Let $M_2 (\mathbb C)$ be the space of complex $2\times 2$
matricies. $A=\begin{bmatrix}a&0\\0&b\end{bmatrix}$. Find for which $a,b \in
\mathbb C$  there exists an open neighborhood $U\in M_2$ of the matrix
$A^5$ and 
a continuously differentiable map $g:U  \rightarrow M_2 (\mathbb C)$
such that  
$g(X)^5=X \forall X\in U$.

 
\end{document}
