How to determine the number of critical points a polynomial scalar field has?

Question

Consider the function

$$f(x,y) = x^3 + 3y - y^3 - 3x$$

How would I be able to determine the number of critical points $f(x,y)$ has?

I know critical points will exist if $\nabla f(x,y) = 0$ or $f_x(x,y) $ and/or $ f_y(x,y)$ don't exist. But if I don't care about the particular values of the critical points, is there just a way to determine the number of critical points?

Thanks in advance!

Answer 1

Using Macaulay2,

Macaulay2, version 1.16
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               LLLBases, MinimalPrimes, PrimaryDecomposition, ReesAlgebra,
               TangentCone, Truncations

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(3*x^2 - 3, 3 - 3*y^2)

              2          2
o2 = ideal (3x  - 3, - 3y  + 3)

o2 : Ideal of R

i3 : dim I

o3 = 0

i4 : degree I

o4 = 4

Hence, there are $4$ critical points. It's easy to conclude that these $4$ points are $(\pm 1, \pm 1)$.

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