I am trying to find the critical points of some functions such as $$f(x, y) = x^4 − x^2y^2 + y^3 − 18x^2 + 3y^2$$ I calculate the gradient, and then find a system of polynomial equations: $$\operatorname{grad}(f(x,y)) = {4x^3-2xy^2-36x\choose-2x^2y+3y^2 + 6y}$$ Our teacher solves it by equaling $x = 0$ and then $y= 0$, which intuitively seems like we're "missing" some solutions. After solving it this way I can't help wondering if I've actually found all the solutions.
I was wondering how many solutions does such a system have in general ($n$ degree polynomials with $m$ variables), and what are some methods to solve them?
$$4x^3-2xy^2-36x= 0 \implies 2x(2x^2-y^2-18)=0$$ $\implies x=0 $
or $2x^2-y^2-18 = 0$ (*)
$$-2x^2y+3y^2 + 6y \implies y(-2x^2+3y+6)=0$$
$\implies y=0 $
or $-2x^2+3y+6 = 0$ (**)
Putting $x=0$ in (**) gives $y=-2$, so one point is $(0,-2)$.
Putting $y=0$ in (*) gives $x=\pm3$, so we get two more points $(3,0)$ and $(-3,0)$.
Both equations are satisified for $x=0,y=0$ so we also get the origin as a point $(0,0)$
To find any more we need to solve (*) and (**) simultaneously. If we can, then along with the others we have found, that should be all the solutions.
From (*), $2x^2=y^2+18$ and substituting this into (**) gives $$-(y^2+18)+3y+6 =0 \implies y^2-3y +12=0$$ which has no real solutions (negative discriminant), so we have found the only 4 points and there are no more.
https://www.wolframalpha.com/input/?i=f(x,+y)+%3D+x%5E4+%E2%88%92+x%5E2y%5E2+%2B+y%5E3+%E2%88%92+18x%5E2+%2B+3y%5E2+stationary+points