Q:

How can I find critical points of the multivariate polynomial? 

I need to find the critical points of the multivariate polynomial and types of critical points 
$f(x_1,x_2,x_3,x_4)=x_2x_3+x_3x_4+x_1x_2+x_1x_2x_3x_4$
What are soft to find the critical points and how I can classification the critical points. 
If taken $$\frac{\partial f}{\partial x_1}=0,\frac{\partial f}{\partial x_2}=0,\frac{\partial f}{\partial x_3}=0,\frac{\partial f}{\partial x_4}=0,$$
then we get
$$x_2+x_2x_3x_4=0$$
$$x_3+x_1+x_1x_3x_4=0$$
$$x_2+x_4+x_1x_2x_4=0$$
$$x_3+x_1x_2x_3=0$$
1.  Do you use a program to solve equations or there is a better way? 

If there is a set of solutions.  How do you critical points ? 

thanks for the help. 

A:

Look at the first equation, there are two possibilities: either $x_2 = 0$, or $1 + x_3 x_4 = 0$.
If $x_2 = 0$, then the remaining equations simplify to
$$\begin{cases} x_2 = 0 \\ x_3 + x_1 + x_3 x_1 x_4 = 0 \\ x_4 = 0 \\ x_3 = 0 ,\end{cases}$$
whence it follows that $x_1 = 0$ and so far we have obtained the solution $(0,0,0,0)$.
If $x_2 \neq 0$ then $x_3 x_4 = -1$, which means that the system simplifies to
$$\begin{cases} x_3 x_4 = -1 \\ x_3 = 0 \\ x_2 + x_4 + x_1 x_2 x_4 = 0 \\ x_3 + x_1 x_2 x_3 = 0 .\end{cases}$$
The first two equations cannot be satisfied simultaneously, therefore this system has no solution. 
We have obtained, then, that $(0,0,0,0)$ is the only critical point of $f$.
The Hessian matrix of $f$ is
$$\begin{pmatrix} 0 & 1 + x_3 x_4 & x_2 x_4 & x_2 x_3 \\ 1 + x_3 x_4 & 0 & 1 + x_1 x_4 & x_1 x_3 \\ x_2 x_4 & 1 + x_1 x_4 & 0 & 1 + x_1 x_2 \\ x_2 x_3 & x_1 x_3 & 1 + x_1 x_2 & 0 \end{pmatrix}$$
which, upon evaluation in $(0,0,0,0)$ produces the matrix
$$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} .$$
Notice that the $1 \times 1$ determinant formed in the upper-left corner is $0$, therefore the matrix is not definite, which shows that $(0,0,0,0)$ is neither a maximum, nor a minimum.  In fact, just noticing that $\dfrac {\partial ^2 f} {\partial x_1 ^2} (0,0,0,0,) = 0$ is enough to deduce this, it isn't really necessary to compute and evaluate the full Hessian.