I have a multivariate polynomial (multivariate polynomial function) $$ R=R_2 R_5+R_2 R_3 R_5+R_4 R_6+R_3 R_4 R_6+R_3 R_5 R_6-R_2 R_3 R_5 R_6-R_2 R_4 R_5 R_6 -R_3 R_4 R_5 R_6+R_2 R_3 R_4 R_5 R_6 $$
The critical points of the multivariate polynomial determine a variety ,by the system $$\frac{\partial R}{\partial R_2}=0,\frac{\partial R}{\partial R_3}=0,\frac{\partial R}{\partial R_4}=0,\frac{\partial R}{\partial R_5}=0,\frac{\partial R}{\partial R_6}=0$$ To solve this system, we can use Mathematica procedures $${Solve}\left[\left\{R_3 R_5-R_3 R_6 R_5+R_3 R_4 R_6 R_5-R_4 R_6 R_5+R_5=0,R_2 R_5-R_2 R_6 R_5+R_2 R_4 R_6 R_5-R_4 R_6 R_5+R_6 R_5+R_4 R_6=0,R_3 R_6-R_2 R_5 R_6+R_2 R_3 R_5 R_6-R_3 R_5 R_6+R_6=0,R_3 R_2-R_3 R_6 R_2+R_3 R_4 R_6 R_2-R_4 R_6 R_2+R_2+R_3 R_6-R_3 R_4 R_6=0,R_3 R_4-R_2 R_5 R_4+R_2 R_3 R_5 R_4-R_3 R_5 R_4+R_4-R_2 R_3 R_5+R_3 R_5=0\right\},\left\{R_2,R_3,R_4,R_5,R_6\right\}\right]$$
The solution should be as follows $$\left\{\left\{R_3\to -1,R_5\to 0,R_6\to 0\right\},\left\{R_3\to -1,R_5\to 0,R_6\to 0\right\},\left\{R_3\to -1,R_5\to 0,R_6\to 0\right\},\left\{R_3\to -1,R_5\to 0,R_6\to 0\right\},\left\{R_2\to 1,R_3\to -1,R_4\to 0,R_5\to 0\right\},\left\{R_2\to \frac{R_4-1}{3 R_4-1},R_3\to 0,R_5\to \frac{3 R_4-1}{R_4-1},R_6\to \frac{1}{R_4}\right\},\left\{R_2\to 0,R_4\to 0,R_5\to 0,R_6\to 0\right\},\left\{R_3\to -1,R_4\to 0,R_5\to 0,R_6\to 0\right\},\left\{R_2\to 0,R_3\to -1,R_4\to 1,R_6\to 0\right\},\left\{R_3\to -1,R_4\to 1,R_5\to 0,R_6\to 0\right\},\left\{R_3\to -1,R_4\to 0,R_5\to 0,R_6\to 0\right\}\right\}$$
For example, $$\{R_3→−1,R_5→0,R_6→0\}$$ In this set $$R_2,R_4$$ not available. What is the explanation for this?
There is more than one solution. Are all of these solutions are considered critical points?
How can I formulate these as critical points?
If you want to find a fixed saddle points (local minima, maxima points ). What is the optimal way? Thanks for the help.
The missing variables may have any value. For instance, ${R_3 \to -1, R_5 \to 0, R_6 \to 0}$ means that $R_2$ and $R_4$ may have any value, i.e. all the points of the form $(R_2, -1, R_4, 0, 0)$ are critical points, regardless of the values of $R_2$ and $R_4$.