<s>[INST] Why does the following nonlinear system have 21 solutions? I am curious as to why the following nonlinear system has 21 solutions (according to Wolfram Alpha). 
$$y+xy^2-x^3+2xz^4=0 \\ -x-y^3-3x^2y+3yz^4=0 \\ -\frac{5}{2}y^2z^3-2x^2z^3-\frac{z^7}{2}=0$$
I see that there is one real and twenty complex solutions:
$(0,0,0)$ 
and

Is there any way to figure out the number of solutions by looking at the degrees of the equations? Like the highest degree of any term is $7$, and there are $3$ variables, so there are $7 \times 3 = 21$ solutions?
 [/INST] Mark the solution $(0,0,0)$ and remove $z^3$ from the last equation. The reduced systems for $z=0$ has two cubic equations, so at most $3⋅3=9$ solutions.
Then $z$ only occurs as $z^4$. Multiply the first equation by $x$, the second by $y$ and express all equations in $a=x^2$, $b=y^2$, $c=xy$ and $d=z^4$, adding the equation $ab-c^2=0$. Then the system reads as
\begin{align}
c+c^2-a^2+2ad&=0 
\\ 
-a-bc-3ac+3cd&=0 
\\ 
-5b-4a-d&=0
\\
ab-c^2&=0
\end{align}
This system has the Bezout bound $8$ for the number of solutions for $(a,b,c,d)$, each of these will have at most $8$ solutions for $(x,y,z)$. This reduces the estimate to $64+9+1=74$ solutions from the $5⋅5⋅7=175$ or $5⋅5⋅4+3⋅3=109$ for the Bezout bound of the (reduced) original system.
For more precise bounds, use the Bernshtejn-Kushnirenko-Khovanskii or BKK bound for sparse polynomial systems. See also the work on the multi-homgeneous resultant.
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